THE 



AMERICAN HOUSE-CARPENTEfi 



A TREATISE 



ON 



THE ART OF BUILDING^ 



AND 



THE STRENGTH OF MATERIALS. 



E. G. HATFIELD, Architect, 

MEM. AM. INST. OF ARCHITECTS. 



SEYBNTH EDITION, REVISED AND ENLARQED 

WITH ADDITIOKAL ILLUSTRATIONS. 



NEW YORK: 

JOHN WILEY & SON, 535 BROADWAY. 

1867. 



k1 



.h! 



"Entered according to Act of Congi*ess, in the year 186T, by 

E. G. HATFIELD, 

In the Clerk's Office of the District Court of the United States, for the Southern District 
of New York. 



By Transfer from 
U.S. Naval Academy 

Aug. 26 1932 







r- 



TABLE OF CONTENTS. 



Intboduotion. — Directions for Drawing 



l-U 



SECTION I.— PEAOTIOAL GEOMETEY 



Definitions .... 
Problems on Lines and Angles 
Problems on the Circle 
Problems on Polygons 



15-70 

71-80 

81-92 

93-106 

Problems on Proportions lOt-110 

Problems on the Conic Sections 111-128 

Demonstrations. — Definitions, Axioms, &c 130-139 

Demonstrations. — Propositions and Corollaries 140-167 

SECTION IL— ABCmTEOTUBE. 



History 

Styles. — Origin, Definitions, Proportions 

Gredan Orders. — ^Doric, Ionic and Corinthian 

Roman Orders. — Doric, Ionic, Corinthian and Composite 

Egjrptian Style 

Buildings generally 

Plans and Elevation for a City Dwelling . . 
Principles of Architecture. — Requisites in a Building 
Principles of Construction. — The Foundations, Column 
Principles of Construction. — The "Wall, Lintel, Arch 
Principles of Construction,— The Yault, Dome, Roof 



168-181 
182-196 
197-211 
212-216 
216, 217 
218-223 
223, 224 
226-229 
230-232 
233-236 
236-238 



VIU TABLE OF CONTENTS. 



SECTION III.— MOULDINGS, C0ENICE8, &o. 

AST& 

Mouldings. — ^Elements, Examples . . . . . 239-250 

Cornices. — Designs 251 

Cornices. — Problems 252-256 



SECTION IV.— FEAMINO, OE CONSTEUCTION. 

First Principles. — Laws of Pressure ....... 25t-282 

Resistance of Materials. — Strength, Stiffness 283-286 

Resistance to Compression. — ^Various kinds 28*7-290 

Results of Experiments on American Materials, Tables I., IE. . . 291-293 

Practical Rules and Examples 294-305 

Resistance to Tension 306 

Results of Experiments on American Materials, Table HI. . . . 307, 308 

Practical Rules and Examples . . 309-316 

Resistance to Cross Strains. — Strength, StifiBiess 311-319 

Resistance to Deflection. — StiShess, Formulse 320-322 

Practical Rules and Examples 323-326 

Table IV.— Weight on Beams, Formulae 326 

Practical Rules and Examples 327-329 

Table V. — Dimensions of Beams, Formulae . . . .... 329 

Resistance to Rupture. — Strength 331 

Results of Experiments on American Materials, Table YL . . . 331 

Table YII.— Safe Weight on Beams, Formulae 333 

Practical Rules and Examples 333 

Table Vm. — Dimensions of Beams, Formulae 334 

Practical Rules and Examples 334 

Systems of Framing, Simplicity of Designs 335 

Floors. — Various, Cross-furring, Reduction of Formulae . . . 336, 337 

Practical Rules and Examples 338-344 

Bridging-strips, Girders, Precautions 345-349 

Pariitiom. — Examples, Load on Partitions, &e 350-353 

Roofs. — Stability, Inclination 354, 355 

Load. — Roofing, Truss, Ceiling, Wind, Snow 356-358 

Strains. — ^Vertical, Oblique, Horizontal 359-368 

Resistance of the Material in Rafter and Tie-beam .... 368 

Dimensions. — Rafter, Braces, Tie-beam, L-on Rods .... 370-374 

Practical Rules and Examples 375-383 

Table IX.— Weight of Roofs, per Foot 376 



TABLE OF CONTENTS. 



IX 



Examples of Roofs 
Problems for Hip-rafter 
Domes. — Examples, Area of Ribs 
Problems in Domes 



Rules for Dimensions . 

Abutments and Piers . 

Stone Bridges, Centreing 

Joints in Timberwork . 

%on "Work. — Pins, Nails, Bolts, Straps 

Iron-Girders. — Cast G-irder, Bow-string, Brick Arch 

Practical Rules and Examples .... 





AEia 


• 


384-386 


• • 


38t-388 


• 


389-391 




392-398 


• 


399-401 


. 


401-405 


. 


406, 407 


. 


408-417 




. 418-427 




428 




. 429-435 




. 431-435 



SECTION v.— DOOES, WINDOWS, &a 



Doors. — Dimensions, Proportions, Examples 
"Windows. — Form, Size, Arrangement, Problems 



436-441 
442-448 



SECTION VL— STAIE8. 



Principles, Pitch Board 

Platform Stairs, Cylinders, Rail, Pace Mould 
"Winding Stairs, Falling Mould, Face Mould, Joints 

Elucidation of Butt Joint 

Quarter-circle Stairs. — ^Falling Mould, Face Mould 

Face Mould. — ^Elucidation 

Face Moulds. — Applied to Plank, Bevils, &c. 
Face Moulds. — ^Another method .... 
Scrolls, Rule, Falling and Face Moulds, Newel Cap 



449-456 
457-463 
469-476 

477 
478-480 

481 
482-484 
485-488 
489-498 



SECTION VII.— SHADOWS. 



Shadows on Mouldings, Curves, Inclinations, && 499-532 

Shadows. — Reflected Light • 523 

APPENDIX. 

PAttB 

Algebraical Signs • • 3 

Trigonometrical Terms . , 5 



X Ti3LE OF CONTENTS. 

Grlossary of Architectural Terms • 1 

Tables of Squares, Cubes and Eoots .••••• .18 

Rules for Beduction of Decimals ,2*1 

Table of Areas and Circumferences of Cirdes .••••• 29 

Table of Capacity of "Wells, Cisterns, &c 83 

Table of Areas of Polygons, &a 34 

Table of Weights of Materials • • .35 



mTRODUCTION. 



Art. 1.— ^A knowledge of the properties and principles of lines 
can best be acquired by practice. Although the various problems 
throughout this work may be understood by inspection, yet they 
will be impressed upon the mind with much greater force, if they 
are actually performed with pencil and paper by the student. 
Science is acquired by study — art by practice : he. therefore, who 
would have any thing more than a theoretical, (which must of 
necessity be a superficial,) knowledge of Carpentry, will attend 
to the following directions, provide himself with the articles here 
specified, and perform all the operations described in the follow- 
ing pages. Many of the problems may appear, at the first read- 
ing, somewhat confused and intricate ; but by making one line 
at a time, according to the explanations, the student will not 
only succeed in copying the figures correctly, but by ordinary 
attention will learn the principles upon which they are based, 
and thus be able to make them available in any unexpected case 
to which they may apply. 

2. — The following articles are necessary for drawing, viz : a 
drawing-board, paper, drawing-pins or mouth-glue, a sponge, a 
T-square, a set-square, two straight-edges, or flat rulers, a lead 
pencil, a piece of india-rubber, a cake of india-ink, a set of draw- 
ing-instruments, and a scale of equal parts. 

3. — The size of the drawing-board must be regulated accord- 
ing to the size of the drawings which are to be made upon it. 
Yet for ordinary practice, in learning to draw, a board about 15 

1 



a AMERICAN HOUSE CARPENTER. 

by 20 inches, and one inch thick, will be found large enoughj 
and more convenient than a larger one. This board should be 
well-seasoned, perfectly square at the corners, and without 
clamps on the ends. A board is better without clamps, because 
the little service they are supposed to render by preventing the 
board from warping, is overbalanced by the consideration that 
the shrinking of the panel leaves the ends of the clamps project- 
ing beyond the edge of the board, and thus interfering with the 
proper working of the stock of the T-square. When the stufi 
is well-seasoned, the warping of the board will be but trifling ; 
and by exposing the rounding side to the fire, or to the sun, it 
may be brought back to its proper shape. 

4. — ^For mere line di-a wings, it is unnecessary to use the hesi 
drawing-paper ; and since, where much is used the expense will 
be considerable, it is desirable for economy to procure paper 
of as low a price as will be suitable for the purpose. The best 
paper is made in England and marked " "Whatman." This is 
a hand-made paper. There is also a machine-made paper at 
about half-price, and the Manilla paper, of various tints of rus- 
set color, is still less in price. These papers are of the various 
sizes needed, and are quite sufficient for ordinary dramngs. 

5. — A drawing-pin is a small brass button, having a steel pin 
projecting from the under side. By having one of these at each 
corner, the paper can be fixed to the board ; but this can be done 
in a much better manner with mouth-glue. The pins will pre- 
vent the paper from changing its position on the board ; but, 
more than this, the glue keeps the paper perfectly tight and 
smooth, thus making it so much the more pleasant to work on. 

To attach the paper with mouth-glue, lay it with the bottom 
side up, on the board ; and with a straight-edge and penknife, 
cut oflT the rough and uneven edge. With a sponge moderately 
wet, rub all the surface of the paper, except a strip around the 
edge about half an inch wide. As soon as the glistening of the 
water disappears, turn the sheet over, and place it upon the 



INTRODUCTION. 



board just where you wish it glued. Conmience upon one of 
the longest sides, and proceed thus : lay a flat ruler upon the 
paper, parallel to the edge, and within a quarter of an inch of it. 
With a knife, or any thing similar, turn up the edge of the papei 
against the edge of the ruler, and put one end of the cake ol 
mouth-glue between your lips to dampen it. Then holdmg it 
upright, rub it against and along the entire edge of the paper 
that is turned up against the ruler, bearing moderately against 
the edge of the ruler, which must be held firmly with the left 
hand. Moisten the glue as often as it becomes dry, until a 
sufficiency of it is rubbed on the edge of the paper. Take 
away the ruler, restore the turned-up edge to the level of the 
board, and lay upon it a strip of pretty stiff" paper. By rubbing 
upon this, not very hard but pretty rapidly, with the thumb nail 
of the right hand, so as to cause a gentle friction, and heat to be 
imparted to the glue that is on the edge of the paper, you will 
make it adhere to the board. The other edges in succession 
must be treated in the same manner. 

Some short distances along one or more of the edges, may 
afterAvards be found loose : if so, the glue must again be applied, 
and the paper rubbed until it adheres. The board must then be 
laid away in a warm or dry place ; and in a short time, the sur- 
face of the paper will be drawn out, perfectly tight and smooth, 
and ready for use. The paper dries best when the board is laid 
level. When the drawing is finished, lay a straight-edge upon 
the paper, and cut it from the board, leaving the glued strip still 
attached. This may afterwards be taken off" by wetting it freely 
with the sponge ; which will soak the glue, and loosen the 
paper. Do this as soon as the drawing is taken off, in order that 
the board may be dry when it is wanted for use again. Care 
must be taken that, in applying the glue, the edge of the paper 
does not become damper than the rest : if it should, the paper 
must be laid aside to dry, (to use at anotlier time,) and anothei 
sheet be used in its place. 



4 AMERICAN HOUSE CARPENTER. 

Sometimes, especially when the drawing board is new, the 
paper will not stick very readily ; but by persevering, this diffi- 
culty may be overcome. In the place of the mouth-glue, a 
strong solution of gum-arabic may be used, and on some 
accounts is to be preferred ; for the edges of the paper need not 
be kept dry, and it adheres more readily. Dissolve the gum in 
a sufficiency of warm water to make it of the consistency of 
linseed oil. It must be applied to the paper with a brush, when 
the edge is turned up against the ruler, as was described for the 
mouth-glue. If two drawing-boards are used, one may be in use 
while the other is laid away to dry ; and as they may be cheaply 
made, it is advisable to have two. The drawing-board having 
a frame around it, commonly called a panel-board, may afford 
rather more facility in attaching the paper when this is of the 
size to suit ; yet it has objections which overbalance that con 
sideration. 

6. — A T-square of mahogany, at once simple in its construc- 
tion, and affording all necessary service, may be thus made. 
Ijet the stock or handle be seven inches long, two and a quarter 
inches wide, and three-eighths of an inch thick: the blade, 
twenty inches long, (exclusive of the stock,) two inches wide, 
and one-eighth of an inch thick. In joining the blade to the 
stock, a very firm and simple joint may be made by dovetailing 
it — as shown at Fig. 1. 




Pig. 1. 



INTRODUCTION. B 

7. — The set-square is in the form of a right-angled triangle ; 
and. is commonly made of mahogany, one-eighth of an inch in 
thickness. The size that is most convenient for general use, is 
six inches and three inches respectively for the sides which con ■ 
tain the right angle ; although a particular length for the sides is 
by no means necessary. Care should be taken to have the square 
corner exactly true. This, as also the T-square and rulers, 
should have a hole bored through them, by which to hang them 
upon a nail when not in use. 

8. — One of the rulers may be about twenty inches long, and 
the other six inches. The pencil ought to be hard enough to 
retain a fine point, and yet not so hard as to leave ineffaceable 
marks. It should be used lightly, so that the extra marks that 
are not needed when the drawing is inked, may be easily rubbed 
off with the rubber. The best kind of india-ink is that which 
will easily rub off upon the plate ; and, when the cake is rub- 
bed against the teeth, will be free from grit. 

9. — The drawing-instruments may be purchased of mathe- 
matical instrument makers at various prices : from one to one 
hundred dollars a set. In choosing a set, remember that the 
lowest price articles are not always the cheapest. A set, com- 
prising a sufficient number of instruments for ordinary use, well 
made and fitted in a mahogany box, may be purchased of the 
mathematical instrument-makers in New York for four or five 
dollars. But for permanent use those which come at ten or 
twelve dollars will be found to be the best. 

10.— The best scale of equal parts for carpenters' use, is one 
that has one-eighth, three-sixteenths, one-fourth, three-eighths, 
one-half, five-eighths, three-fourths, and seven-eighths of an 
inch, and one inch, severally divided into twelfths, instead ot 
being divided, as they usually are, into tenths. By this, if it be 
required to proportion a drawing so that every foot of the object 
represented will upon the paper measure one-fourth of an inch, 
use that part of the scale which is divided into one-fourths of an 



6 AMERICAN HOUSE-CARPENTER. 

inch^ taking for every foot one of those divisions, and for evei^ 
inch one of the subdivisions into twelfths ; and proceed in like 
manner in proportioning a drawing to any of the other divisions 
of the scale. An instrument in the form of a semi-circle, called a 
protractor^ and used for laying down and measuring angles, is 
of much service to surveyors, but not much to carpenters. 

11. — In drawing parallel lines, when they are to be parallel 
to either side of the board, use the T-square; but when it is 
required to draw lines parallel to a line which is drawn in a 
direction oblique to either side of the board, the set-square must 
be used. Let a 6, {Fig. 2,) be a line, parallel to which it is 




Fig 2. 



desired to draw one or more lines. Place any edge, as c d, ol 
the set-square even with said line ; then place the ruler, g h, 
against one of the other sides, as c e, and hold it firmly ; slide 
the set-square along the edge of the ruler as far as it is desired, 
as at/; and a line drawn by the edge, i/, will be parallel to a h. 
12. — To draw a line, as k I, {Fig. 3,) perpendicular to another, 
as a 5, set the shortest edge of the set-square at the line, a b; 
place the ruler against the longest side, (the hypothenuse of the 
right-angled triangle ;) hold the ruler firmly, and slide the set- 
square along until the side, e d touches the point, k ; then the 
line, I kj drawn by it, will be perpendicular to a 6. In like 



INTRODUCTION. 



manner, the drawing of other problems may be facilitated, as will 
be discovered in using the instruments. 




Fig. 3. 



13. — In drawing a problem, proceed, with the pencil sharpened 
to a point, to lay down the several lines until the whole figure is 
completed ; observing to let the lines cross each other at the 
several angles, instead of merely meeting. By this, the length 
of every line will be clearly defined. With a drop or two of 
vater, rub one end of the cake of ink upon a plate or saucer, 
until a sufliciency adheres to it. Be careful to dry the cake of 
ink ; because if it is left wet, it will crack and crumble in pieces. 
With an inferior camePs-hair pencil, add a little water to the 
ink that was rubbed on the plate, and mix it well. It should be 
diluted sufficiently to flow freely from the pen, and yet be thick 
enough to make a black line. With the hair pencil, place a 
little of the ink between the nibs of the drawing-pen, and screw 
the nibs together until the pen makes a fine line. Beginning 
with the curved lines, proceed to ink all the lines of the figure ; 
being careful now to make every line of its requisite length. It 
they are a trifle too short or too long, the drawing will have a 
ragged appearance; and this is opposed to that neatness and 
accuracy which is indispensable to a good drawing. When the 
ink is dry, eflface the pencil-marks with the india-rubber. If 



8 - AMERICAN HOUSE-CARPENTER. 

the pencil is used lightly, they will all rub off, leaving those 
lines only that were inked. 

14. — ^In problems, all auxiliary lines are drawn light ; while 
the lines given and those sought, in order to be distinguished at 
a glance, are made much heavier. The heavy lines are made 
so, by passing over them a second time, having the nibs of the 
pen separated far enough to make the lines as heavy as desired. 
If the heavy lines are made before the drawing is cleaned with 
the rubber, they will not appear so black and neat ; because the 
india-rubber takes away part of the ink. If the drawing is a 
ground-plan or elevation of a house, the shade-lines, as they are 
termed, should not be put in until the drawing is shaded ; as 
there is danger of the heavy lines spreading, when the brush, in 
shading or coloring, passes over them. If the lines are inked 
with common writing-ink, they will, however fine they may be 
made, be subject to the same evil ; for which reason, india-ink 
is the only kind to be used. 



THE 

AMERICAN HOUSE-CARPENTER, 



SECTION I.— PRACTICAL GEOMETRY. 



DEFINITIONS. 



15. — Geometry treats of the properties of magnitudes^ 

16. — A point has neither length, breadth, nor thickness. 

17. —A, line has length only. 

18. — Superficies has length and breadth only. 

19. — A plane is a surface, perfectly straight and even in every 
direction ; as the face of a panel when not warped nor winding. 

20. — A solid has length, breadth and thickness. 

21. — A rights or straight^ line is the shortest that can be 
drawn between two points. 

22. — Parallel lines are equi-distant throughout their length. 

23. — An angle is the inclination of two lines towards one 
anothcjr. {Fig. 4.) 




Fig. 4. Fig. 5. Pig. & 



10 



AMERICAN HOUSE-CARPZNTER. 



24. — A right angle has one line perpendicular to the othei. 
{Fig. 5.) 

25. — An oblique angle is either greater or less than a right 
angle. {Fig. 4 and 6.) 

26. — An acute angle is less than a right angle. {Fig- 4.) 

27. — An obtuse angle is greater than a right angle. {Fig. 6.) 

When an angle is denoted by three letters, the middle one, in 
the order they stand, denotes the angular point, and the other 
two the sides containing the angle ; thus, let a 6 c, {Fig. 4,) be 
the angle, then b will be the angular point, and a b and b c will 
De the two sides containing that angle. 

28. — A triangle is a superficies having three sides and angles, 

(Ft^. 7, 8, 9 and 10.) 





Fig. 8. 



29. — An equi-lateral triangle has its three sides equal. 
(Ftg.7.) 

30. — An isosceles triangle has only two sides equal. {Fig. 8.) 
31. — A scalene triangle has all its sides unequal. (Fig. 9) 





Fig. 9. 



Fi?. 10. 



32. — A right-angled triangle has one right angle. {Fig. 10.) 

33. — An acute-angled triangle has all its angles acute. 
{Fig. 7 and 8.) 

34. — An obtuse-angled triangle has one obtuse angle. 
{Fig. 9.) 

35.— A quadrangle has four sides and four angles, (i^. 11 
to 16.) 



PRACTICAL GEOMETRY. 



11 



Fig. 11. 



Fiff. 12. 



36. — A parallelogram is a quadrangle having its opp()sit€ 
sides parallel. [Fig, 11 to 14.) 

37. — A rectangle is a parallelogram, its angles being right 
angles. [Fig. 11 and 12.) 

38. — A square is a rectangle having equal sides. [Fig. 11.) 

39. — A rhombus is an equi-lateral parallelogram having ob- 
Uque angles. [Fig. 13.) 




7 



Fig. 13. 



Fig. 14. 



40. — A rhomboid is a parallelogram having oblique angles. 
(Fig. 14.) 
41. — A trapezoid is a quadrangle having only two of its sides 

parallel. [Fig. 15.) 




Fig. 15. 



Fig. 16. 



42. — A trapezium is a quadrangle which has no two of its 

sides parallel. [Fig. 16.) 

43. — A 'polygon is a figure bounded by right lines. 

44. — A regular polygon has its sides and angles equal. 

45. — An irregular polygon has its sides and angles unequal. 

46. — A trigon is a polygon of three sides, [Fig. 7 to 10 ;) 
a tetragon has four sides, [Fig. 11 to 16 ;) a pentagon has 



12 



AMERICAN HOUSE-CARPENTER. 



five, {Fig. 17 ;) a hexagon six, \Fig. 18 ;) a heptagon seven, 
[Fig. 19 ;) an octagon eight, [Fig. 20 ;) a nonagon nine ; a 
decagon ten ; an undecagon eleven ; and a dodecas^on twelve 
sides. 






Fig. 17. 



Fig. 18. 



Fig. 19. 



Fig. 20. 



47. — A cir^cle is a figure bounded by a curved line, called the 
circumference ; which is every where equi-distant from a cer- 
tain point within, called its centre. 

The circumference is also called the periphery/, and sometimes 
the circle. 

48. — The radius of a circle is a right line drawn from the 

centre to any point in the circumference, [a b, Fig. 21.) 

All the radii of a circle are equal. 




Fig. 21. 



49. — The diameter is a right line passing through the centre, 
and terminating at two opposite points in the circumference. 
Hence it is twice the length of the radius, (c d, Fig. 21.) 

50. — An arc of a circle is a part of the circumference, (c b or 
bed, Fig. 21.) 

51. — A chord is a right line joining the extremities of an arc. 
ihd, Fig. 21.) 



PRACTICAL GEOMETRY. 



13 



52. — A segment is any part of a circle bounded by an arc and 
its chord. (.4, Fig. 21.) 

53. — A sector is any part of a circle bounded by an arc and 
two radiij drawn to its extremities. (J5, Fig. 21.) 

54. — A quadrant^ or quarter of a circle, is a sector having a 
quarter of the circumference for its arc. (C, Fig. 21.) 

55. — A tangent is a right line, which in passing a curve, 
touches, without cutting it. [f g-, Fig. 21.) 

56. — A cone is a solid figure standing upon a circular base 
diminishing in straight lines to a point at the top, called its 
vertex. {Fig. 22.) 




Fig. 22. 




57. — The axis of a cone is a right line passmg through it, froin 
the vertex to the centre of the circle at the base. 

58. — An ellipsis is described if a cone be cut by a plane, not 
parallel to its base, passing quite through the curved surface, 
(a 6, Fig. 23.) 

59. — A parabola is described if a cone be cut by a plane, 
parallel to a plane touching the curved surface, (c cZ, Fig. 23 — 
c d being parallel tofg.) 

60. — An hyperbola is described if a cone be cut by a plane^ 
parallel to any plane within the cone that passes through its 
vertex, (e A, Fig. 23.) 

61. — Foci are the points at which the pins are placed in de* 
scribing an ellipse. (See Art. 115, and/, /, Fig. 24.) 



u 



AMERICAN HOUSE-CARPENTER. 




62. — The transverse axis is the longest diameter of the 
ellipsis, {a 6, Fig. 24.) 

63. — The co7ijiigate axis is the shortest diameter of the 
ellipsis ; and is, therefore, at right angles to the transverse axis, 
(c d, Fig. 24.) 

64. — The parameter is a right line passing through the focus 
of an ellipsis, at right angles to the transverse axis, and termina- 
ted by the curve, {g h and g t^ Fig. 24.) 

65. — A diameter of an ellipsis is any right line passing 
through the centre, and terminated by the curve, [k /, or m ?t, 
Fig. 24.) 

66. — A diameter is conjugate to another when it is parallel to 
a tangent drawn at the extremity of that other — thus, the diame- 
ter, m w, [Fig. 24,) being parallel to the tangent, o p, is therefore 
conjugate to the diameter, k I. 

67. — A double ordinate is any right line, crossing a diameter 
of an ellipsis, and drawn parallel to a tangent at the extremity of 
that diameter, (i t^ Fig. 24.) 

68. — A cylinder is a solid generated by the revolution of a 
right-angled parallelogram, or rectangle, about one of its sides ; 
and consequently the ends of the cylinder are equal circles. 
{Fig.2o.) 



PRACTICAL GEOMETRY. 



15 



Fig, 25. 




Fig. 26. 



69. — The axis of a cylinder is a right hne passing through it, 
from the centres of the two circles which form the ends. 

70. — A segment of a cylinder is comprehended under three 
planes, and the curved surface of the cylinder. Two of these 
are segments of circles : the other plane is a parallelogram, called 
by way of distinction, the plane of the segment. The circular 
segments are called, the ends of the cylinder. {Fig. 26.) 



J\^. B. — For Algebraical Signs, Trigonometrical Terms, &c.j 
see Appendix. 



PROBLEMS 



RIGHT LINES AND ANGLES. 



71. — 2^0 bisect a line. Upon the ends of the line, a b, {Fig. 
27,) as centres, with any distance for radius greater than hall 





ct 6j describe arcs cutting each other in c and d ; draw the line, 

c d, and the point, e, where it cuts a 6, will be the middle of the 

line, a b. 

In practice, a line is generally divided with the compasses, oi 
dividers ; but this problem is useful where it is desired to draw, 
at the middle of another line, one at right angles to it. (See 
Art. 85.) 

d 




Fi-. 28. 



72. — To erect a perpendicular. From the point, a, {Fig' 28,) 



PRACTICAL GEOMETRY. 



17 



set ofi any distance, as a b, and the same distance from a to c , 
upon c, as a centre, with any distance for radius greater than c a, 
describe an arc at d ; upon 6, with the same radius, describe 
another at d ; join d and a, and the line, d a, will be the per- 
pendicular required. 

This, and the three following problems, are more easily per- 
formed by the use of the set-square — (see Art. 12.) Yet they 
are useful when the operation is so large that a set-square cannot 
be used. 



V 


d 




e^*v^^^^_^ 


r 


■^'f 



Fig. 29. 

73. — To let fall a perpendicular. Let a, [Fig. 29,) be the 
point, above the line, h c, from which the perpendicular is re- 
quired to fall. Upon a, with any radius greater than a c?, de- 
scribe an arc, cutting 6 c at e and/; upon the points, e and/ 
with any radius greater than e d, describe arcs, cutting each 
other at g ; join a and ^, and the line, a d, will be the perpen- 
dicular required. 




Fig. 30. 

74. — To erect a perpendicular at the end of a line. Let &, 
{Fig. 30,) at the end of the line, c a, be the point at which the 
perpendicular is to be erected. Take any point, as 6, above tlie 
line, c a, and with the radius, h a, describe the arc, d a e. 
through d and 6, draw the line, d e ; join e and a, then e a will 
be the perpendicular required. „ 



18 



AMERICAN HOUSE-CARPENTER. 



The principle here made use of, is a very important one ; and 
is appHed in many other cases — (see Art. 81, 6, and Art, 84. 
For proof of its correctness, see Art. 156.) 




74, a, — A second method. Let b, {Fig. 31 ,) at the end of the 

line, a b, be the point at which it is required to erect a perpen- 
dicular. Upon 6, with any radius less than b a, describe the arc, 
c e d ; upon c, with the same radius, describe the small arc at e, 
and upon e, another at d ; upon e and d, with the same or any 
other radius greater than half e d, describe arcs intersecting at/; 
join /and 6, and the line,/ 6, will be the perpendicular required. 
This method of erecting a perpendicular and that of the fol- 
lowing article, depend for accuracy upon the fact that the side 
of a hexagon is equal to the radius of the circumscribing circle. 




d 
Fig. 32. 



74, b. — A third method. Let b, {Fig. 32,) be the given point 
at which it is required to erect a perpendicular. Upon b^ with any 
radius less than b <i, describe the quadrant, d ef; upon rf, with 
the same radius, describe an arc at e, and upon c, another at c , 



PRACTICAL GEOMETRY. 19 

through d and e, draw d c, cutting the arc in c ; jcin c and 6, 
then c h will be the perpendicular required. 

This problem can be solved by the six^ eight and ten rule, 
as it is called ; which is founded upon the same principle as 
the problems at Art. 103, 104 ; and is applied as follows. Let 
a 0?, [Fig. 30,) equal eight, and a e, six ; then, \i d e equals ten, 
the angle, c a c?, is a right angle. Because the square of six 
and that of eight, added together, equal the square of ten, thus : 
6 X 6 = 36, and 8 X 8 = 64 ; 36 + 64 = 100, and 10 x 10 = 
100. Any sizes, taken in the same proportion, as six, eight and 
ten, will produce the same effect : as 3, 4 and 5, or 12, 16 and 
20. (See Art. 103.) 

By the process shown at Fig. 30, the end of a board may be 
squared without a carpenters'-square. All that is necessary is a 
pair of compasses and a ruler. Let c a be the edge of the board, 
and a the point at which it is required to be squared. Take the 
point, 6, as near as possible at an angle of forty-five degrees, or on 
a mi^re-line, from a, and at about the middle of the board. This 
is not necessary to the working of the problem, nor does it affect 
its accuracy, but the result is more easily obtained. Stretch the 
compasses from h to a, and then bring the leg at a around to d ; 
draw a line from c?, through ft, out indefinitely ; take the dis- 
tance, d b, and place it from b to e ; join e and a ; then e a will 
be at right angles to c a. In squaring the foundation of a build- 
ing, or laying-out a garden, a rod and chalk-line may be used 
instead of compasses and ruler. 

7^. — To let fall a perpendicular near the end of a line. 

Let c, {Fig. 30,) be the point above the line, c a, from which the 

perpendicular is required to fall. From e, draw any line, as e c?, 

obliquely to the line, c a ; • bisect e d at b ; upon 6, with the 

radius, b e, describe the arc, e a d ; join e and a ; then e a will 

be the perpendicular required. 




76. — To make an angle, (as e df Fig. 33,) equal to a given 
angle, (as b a c.) From the angular point, a, with any radius 
describe the arc, he; and with the same radius, on the line, d e, 



20 



AMERICAN H0USE-CARPB:NTER. 



and from the point, d^ describe the arc, fg- ; take the distance, 

b c, and upon gj describe the small arc at/; join /and d ; and 

the angle, e df, will be equal to the angle, b a c. 

If the given line upon which the angle is to be made, is situa- 
ted parallel to the similar line of the given angle, this may be 
performed more readily with the set-square. (See Art. 11.) 




Fig. 34. 



77. — To bisect an angle. Let a b c, {Fig. 34,) be the angle 

to be bisected. Upon b, with any radius, describe the arc, a c ; 

upon a and c, with a radius greater than half a c, describe arcs 

cutting each other at d ; join b and d ; and b d will bisect the 

angle, a b c, us was required. 

This problem is frequently made use of in solving other pro- 
blems ; it should therefore be well impressed upon the meniory. 




Fig. 35. 

78. — To trisect a right angle. Upon a, {Fig. 35,) with any 

radius, describe the arc, b c ; upon b and c, with the same radius, 

describe arcs cutting the arc, b c^aX d and e ; from d and e, draw 

lines to a, and they will trisect the angle as was required. 

The truth of this is made evident by the following operation. 
Divide a circle into quadrants : also, take the radius in the divi- 
ders, and space off the circumference. This will divide the 
circumference into just six parts. A semi-circumference, there- 



PRACTICAL GEOMETRY. 21 

fore, IS equal to three, and a quadrant to one and a half of those 
parts. The radius, therefore, is equal to | of a quadrant ; and 
this is equal to a right angle. 



Fig. 36. 

79. — Through a given pointy to draw a line parallel to a 
given line. Let a, i^Fig. 36,) be the given point, and b c the 
given line. Upon any point, as d^ in the line, h c, with the 
radius, d a, describe the arc, a c; upon a, with the same radius, 
describe the arc, d e ; make d e equal to a c ; through e and a 
draw the line, e a ; which will be the line required. 

This is upon the same principle as Art. 76. 




80. — To divide a given line into any number of equal paYts. 

Let a -6, {Fig. 37,) be the given line, and 5 the number of parts. 

Draw a c, at any angle \o ab ; on a c, from a, set off 5 equal 

parts of any length, as at 1, 2, 3, 4 and c ; join c and b ; through 

the points, 1, 2, 3 and 4, draw 1 e,2f,3g and 4 h, parallel to 

c b ; which will divide the line, a b, as was required. 

The lines, a b and a c, are divided in the same proportion. 
(See Art. 109.) 

THE CIRCLE. 

8L — To find the centre of a circle. Draw any chord, as a ^, 



22 



4MERICAN HOUSE-CARPENTER. 




(Fig. 38,) and bisect it Avith the perpendicular, c d ; tisect c U 
with the line, e/, as at g ; then g is the centre as was required. 




81, a. — A second method. Upon any two points in the cir- 
cumference nearly opposite, as a and h, (Fig. 39,) describe arcs 
cutting each other at c and d : take any other two ]^oints, as e 
and/, and describe arcs intersecting as at g and h ; join g and A, 
and c and d ; the intersection, o, is the centre. 

This is upon the same principle as Art. 85. 




Fii^. 4a 



81, b*-^A third method. Draw any chord, as a b, {Fig» 40,) 



PRACTICAL GEOMETRY. 2,*^ 

and from the point, a, draw a c, at right angles to a b ; join 

c and b ; bisect c 6 at d — which will be the centre of the circle. 

If a circle be not too large for the purpose, its centre may very 
readily be ascertained by the help of a carpenters'-square, thus : 
app' y the corner of the square to any point in the circumference, 
as at a ; by the edges of the square, (which the lines, a b and 
a c, represent,) draw lines cutting the circle, as at b and c ; join 
b and c ; then if 6 c is bisected, as at c?, the point, d, will be the 
centre. (See Art. 156.) 




fig. 41. 



82. — At a given point in a circle, to draw a tangent thereto. 
Let «, {Fig. 41,) be the given point, and b the centre of the cir- 
cle. Join a and b ; through the point, a, and at right angles to 
a b, draw c d ; then cdis the tangent required. 




Fifir.42 

^3. — The same, loithout m^aking use of the centre of the 
circle. Let a, {Fig. 42,) be the given point. From a, set off 
any distance to b, and the same from b to c ; join a and c , 
upon a, with a b for radius, describe the arc, d b e ; make d b 
equal to be; through a and d, draw a line ; this will be the 
tangent required. 

The correctness of this method depends upon the fact that 
the angle formed by a chord and tangent is equal to any 



24 



AMEEICAX HOrSE-CAEPEXTEK. 



inscribed angle in the opposite segment of the cu'cle, {Art 
163 ;) ah being the chord, and lea the angle in the opposite 
segment of the circle, l^ow, the angles d ah and h c a ai-e 
equal, because the angles d ah and h a c are, by construction, 
equal ; and the angles h a c and h c a are equal, because the 
triangle ah c is slu isosceles triangle, having its two sides, a h 
and h c, by construction equal ; therefore the angles d ah and 
h G a are equal. 




Firr. 43. 



84. — A circle and a tangent given^ to find the point of cortr 

tact. From any point, as a, {Fig. 43,) in the tangent, h c. draw 

a line to the centre d / bisect a d at e / upon e, with the radius, 

e a, describe the arc, afd; f is the point of contact required. 

If/ and d were joined, the line would form right angles with 
the tangent, h c. (See Art. 156.) 




Fig. 44. 



85. — Through any three points not in a straight li7ie, to draw 
a circle. Let «, h and c, {Fig. 44,) be the three given points. 
Upon a and 5, with any radius greater than half a h, describe 



PKACTICAL GEOMETET. 



25 



arcs intersecting a ': d and e / upon h and c, wita any radius 
greater than half h e, describe arcs intersecting at / and g ; 
through d and e, draw a right line, also another through / and 
g / upon the intersection, h, with the radius, h a, describe the 
circle, ah c, and it will be the one required. 




Fig. 45. 



86. — Three points not in a st/raigJit line heing given, tojmd 

a fourth that shall, with the three, lie in the circumference of a 

circle. Let ate, {Fig. 45,) be the given points. Connect 

them with right lines, forming the triangle, a oh j bisect the 

angle, da, {Art. T7,) with the line h d ; also bisect came, 

and erect e d, perpendicular to a c, cutting h din d; then d is 

the fourth point required. 

A fifth point may be found, as at/*, by assuming a, d and 5, 
as the three given points, and proceeding as before. So, also, 
any number of points may be found ; simply by using any three 
already found. This problem will be serviceable in obtaining 
short pieces of very flat sweeps. (See Art. 397.) 

The proof of the correctness of this method is found in the 

fact that equal chords subtend equal angles, {Art. 162.) Join 

d and c ; then since a e and e c are, by construction, equal, 

therefore the chords a d and d c are equal ; hence the angles 

they subtend, d h a and dh c, are equal. So likewise chords 

drawn from a tof and from /" to d, are equal, and subtend the 

equal angles, dhf and fh a. Additional points, hey and a or 

h, may be obtained on the same principle. To obtain a point 

beyond a, on h, as a centre, describe with any radius the arc 

ion, make o n equal to oi, through h and n draw 5 ^ / on « as 



26 



A3IEEICAN HOUSE-CAKPENTEE. 



centre and with af for radius, describe the arc, cutting gh 2X 
g^ then g is the point sought. 




87.— T<9 describe a segment of a circle hy a set-triangle. Let 
a 5, (Fig. 46,) be the chord, and c d the height of the segment. 
Secure two straight-edges, or rulers, in the position, c e and cf, 
by nailing them together at c, and affixing a brace from e to 
// put in pins at a and h / move the angular point, c, in the 
direction, a ch ; keeping the edges of the triangle hard against 
the pins, a and 5 / a pencil held at c will describe the arc, a ch. 

A curve described by this process is accurately circular^ and 

is not a mere approximation to a circular arc, as some may 

suppose. This method produces a circular curve, because all 

inscribed angles on one side of a chord line are equal. {Art. 

161.) To obtain the radius from a chord and its versed sine, 

see Art. 165. 

If the angle formed by the rulers at c be a right angle, the 
segment described will be a semi-circle. This problem is use- 
ful in describing centres for brick arches, when they are re- 
quired to be rather flat. Also, for the head hanging-stile of a 
window-frame, where a brick arch, instead of a stone Hntel, is 
to be placed over it. 

87 a. — To find the radius of an arc of a circle when the 

chord and versed sine are given. The radius is equal to the 

sum of the squares of half the chord and of the versed sine, 

divided by twice the versed sine. This is expressed, algebraic- 

(gf+v" 
ally, thus — ^=="^"0 > where r is the radius, c the chord, and v 

the versed sine. {Art. 165.) 

Example. — In a given arc of a circle, a chord of 12 feet hag 



PEACTicAL geo:metey. 27 

the rise at tlie middle, or the versed sine, equal to 2 feet, what 

is the radius ? 

Half the chord equals 6, the square of 6 is, 6 X 6 = 36 
The square of the versed sine is, 2 x 2 = 4 

Their sum equals, 40 

Twice the versed sine equals 4, and 40 divided by 4 equals 10 
Therefore the radius, in this case, is 10 feet. This result it 
shown in less space and more neatly bj^ using the above alge- 
braical formula. For the letters, substituting their value, the 

formula r = -^^^P- — becomes r = ^^ — -— , and performing 

2 V 2x2 r fc 

the arithmetical operations here indicated equals 
6^ + 2' _ 36 + 4 _ 40 _ 

4 "" 4 "~ 4 "" 

87 b. — To jmd the versed sine of cm arc of a circle when the 

radius and chord are given. The versed sine is equal to the 

radius, less the square root of the difference of the squares of 

the radius and half chord : expressed algebraically thus — v = r 

— \^ r"" — (l)^, where r is the radius, v the versed sine, and e 

the chord. {Art. 158.) 

Exarrvple. — In an arc of a circle whose radius is Y5 feet, 
what is the versed sine to a chord of 120 feet ? By the table 
in the Appendix it will be seen that — 

The square of the radius, 75, equals, , . 5625 
The square of half the chord, 60, equals, . 3600 



The difference is, 2025 



The square root of this is, .... 46 
This deducted from the radius, ... 75 



The remainder is the versed sine, — 30 

This is expressed by the formula thus — 
V z= 75 - V75'-(.-4^y=75 — V 5625 - 3600 = 75 — 45 = 30 




28 



AMERICAN HOUSE-CAICPEI^TEE. 



88. — To describe the aegment of a eirde hy intersection of 

lines. Let a 5, {Fig. 47,) be the cliord, and c d the height of 

the segment. Through c, draw e f parallel to ah ; draw h f 

at right angles to ch ; make c e equal to c f ; draw a g and 

h A, at right angles to a h ; divide c e., cf d a^dh^a g^ and 

h A, each into a like number of equal parts, as four ; draw the 

lines, 1 1, 2 2, &c., and from the points, o, o and 6», draw lines 

to G / at the intersection of these lines, trace the curve, (Z c 5, 

which will be the segment required. 

In very large work, or in laying out ornamented gardens, 
&c., this will be found useful ; and where the centre of the 
proposed arc of a circle is inaccessible it will be invaluable. 
(To trace the cm-ve, see note at Art. 117.) 

The lines e a, c d and f 5, would, were they extended, meet 

in a point, and that point would be in the opposite side of the 

circumference of the circle of which a chh a segment. Tlie 

lines 1 1, 2 2, 3 3, would likewise, if extended, meet in the 

same point. The line, c d^ if extended to the opposite side of 

the circle, would become a diameter. The line,/" 5, forms, by 

construction, a right angle with h c, and hence the extension of 

f h would also form a right angle with h c, on the ojDposite side 

oi h c J and this right angle would be the inscribed angle in 

the semicircle ; and since this is required to be a right angle, 

{Art. 156,) therefore the construction thus far is correct, and it 

will be found likewise that at each point in the curve formed 

by the intersection of the radiating lines, these intersecting 

lines are at right angles. 





a 

Fig. 47 a. 



88 a. — Points in the circumference of a circle may be ob- 
tained arithmetically, and positively accurate, by the calcula- 
tion of ordinates^ or the parallel lines, 1, 2, 3, 4. {Fig. 



PEACTICAL GEOlVrETET. 29 

4:7 a.) These ordinates are drawn at right angles to the chord 
line, a 5, and they may be drawn at any distance apart, either 
equally distant or unequally, and there may be as many of 
them as is desirable ; the more there are the more points in 
the curve will be obtained. If they are located in pairs, 
equally distant from the versed sine, c d^ calculation need be 
made only for those on one side of g d^ as those on the opposite 
side will be of equal lengths, respectively ; for example, 1, on 
the left-hand side of c d^ is equal to 1 on the right-hand side, 
2 on the right equals 2 on the left, and in like manner for 
the others. 

The length of any ordinate is equal to the square root of 
the difference of the squares of the radius and abscissa, less 
the difference between the radius and versed sine. {Art. 166.) 
The abscissa being the distance from the foot of the versed sine 



to the foot of the ordinate. Algebraically, y = Vr^— sg"^ — 
{r — -y), where y is put to represent the ordinate ; a?,, the ab- 
scissa ; v^ the versed sine ; and r, the radius. 

Example. — An arc of a circle has its chord, a h, {Fig. 47 <2,) 
100 feet long, and its versed sine, c d, ^ feet. It is required to 
ascertain the length of ordinates for a sufficient number of 
points through which to describe the curve. To this end it is 
requisite, first, to ascertain the radius. This is readily done in 

accordance with Art. 87 a. For, ^^ , becomes ^^ — = 

2-^ 2x5 

252'5 = radius. Having the radius, the curve might at once 

be described without the ordinate points, but for the imjDracti- 

cability that usually occurs, in large, flat segments of the circle, 

of getting a location for the centre ; the centre usually being 

inaccessible. Tlie ordinates are, therefore, to be calculated. 

In Fig. 47 a the ordinates are located equidistant, and are 10 

feet apart. It will only be requisite, therefore, to calculate 

those on one side of the versed sine, c d. For the first ordinate, 

1, the formula, y = ^/r'' — x' — {r — v) becomes 

y = V252-5^ - 10"^ - (2 52-5 - 5). 

= V'63756-25 - 100 - 247-5. 

= 252-3019 - 247-5. 

= 4-8019 =: the first ordinate, 1. 



30 



AM1.EICA1T HOUSE -CAKPENTEE. 



For tJie becond — ■ 



y = V252-5' - 20^ -(252*5 - 5). 
= 251-7066 - 247-5. 
= 4-2066 = the second ordinate, 02. 

For the thi rd — 

y = V252-5-^ - 30=^ - 247*5. 
= 250-7115 - 247-5. 
= 3-2115 = the third ordinate, 03. 
For the fourth — 

y = V252-5'' - 40' - 247-5. 
= 249-3115 - 247-5. 
= 1-8115 = the fourth ordinate, 04. 

The results here obtained are in feet and decimals of a foot. 
To reduce these to feet, inches, and eighths of an inch, proceed 
as at Reduction of Decimals in the Appendix. If the two-feet 
rule, used by carpenters and others, were decimally divided, 
there would be no necessity of this reduction, and it is to be 
hoped that the rule will yet be thus divided, as such a reform 
would much lessen the labor of computations, and insure more 
accurate measurements. 
Yersed sine, c ^, = ft. 5*0 
Ordinates, 01,= 4-8019 
" 2,= 4-2066 

" 3,= 3-2115 

" 4,= 1-8115 



ft. 5-0 inches. 

4-9f inches nearly. 
4-2i inches nearly. 
3-22 inches nearly. 
1-91 inches nearly. 




Fig. 48. 



89. — In a given cmgle, to describe a ta/nged curve. Let ah c, 

{Fig. 48,) be the given angle, and 1 in the line, a h, and 5 in 

the line, h c, the termination of the curve. Divide 1 h and b 5 

into a like number of equal parts, as at 1, 2, 3, 4 and 5 ; join 1 

and 1, 2 and 2, 3 and 3, &c. ; and a regular curve will be 

formed that will be tangical to the line, a h, at tha point, 1, and 

to 5 c at 5. 

This is of much use in stair-building, in easing the angles 
formed between the wall-string and the base of the hall, also 



PRACTICAL GEOMETET. 



31 



between the front sti-ing and level facia, and in many other 
instances. The curve is not circular, but of the form of the 
parabola, {Fig. 93 ;) yet in large angles the difference is not 
perceptible. This problem can be applied to describing the 




curve for door heads, window-heads, &c., to rather better ad- 
vantage than Art. 87. For instance, let a 5, {Fig- 49^ be the 
width of the opening, and c d the height of the arc. Extend g 
d^ and make d e equal to c d ; join a and e, also e and h / and 
proceed as directed above. 




Tig. 50 



90. — To describe a circle withvn cmy given tnricmgle^ so that 
the sides of the triangle shall he tam^gical. Let ah c^ {Fig. 50,) 
be the given triangle. Bisect the angles a and 5, according to 
Art 77 ; upon d', the point of intersection of the bisecting lines, 
with the radius, d e, describe the required circle. 




32 



AMEKICAN HOrSE-CAErEXTEE. 



91. — About a given circle, to describe an equi-lateral tri- 
angle. Let a db c, {Fig, 51,) be the given circle. Draw the 
diameter, c d ; upon d, with the radius of the given circle, 
describe the arc, aeb ; join a and b ; draw/ g, at right angles 
to d c ; make/* c and c g, each equal to ab ; from / through 
a, di-aw/ A, also from g, through b, draw g h; then fg h will 
be the triangle requii-ed. 




92. — To fold a right line nearly equal to the circumference 
of a circle. Let abed, {Fig. 52,) be the given circle. Draw 
the diameter, a c ; on this erect an equi-lateral triangle, a e c, 
according to Art. 93 ; draw g f parallel to a c ; extend e c to 
/ also e a to g ; then g f will be nearly the length of the 
semi-circle, a d c j and twice g f will nearly equal the circum- 
ference of the circle, ab c d, ^^ was required. 

Lines drawn from e, through any points in the circle, as o, o 
and 6>, to ;p, p and p, will divide g f in the same way as the 
semi-circle, a d c,\^ divided. So, any portion of a circle may 
be transferred to a straight line. This is a very useful pro- 
blem, and should be well studied ; as it is frequently used to 
solve problems on stairs, domes, &c. 

92, a. — Another method. Let a bf c, {Fig. 53,) be the given 

circle. Draw the diameter, ac ; from d, the centre, and at right 

angles to a c, draw db ; join b and c ; bisect b c 2it e ; from d, 

through e, draw df ; then ^/ added to three times the diameter, 

will equal the circumference of the circle sufficiently near for 



PEACTICAL GEOMETRY. ?,3 




Fig. 53. 



many uses. The result is a trifle too large, If the circumfer- 
erence found by this rule, be divided by 648*22, the quotient 
will be the excess. Deduct this excess, and the remainder 
will be the true circumference. This problem is rather more 
curious than useful, as it is less labor to perform the operation 
arithmetically: simply multiplying the given diameter by 
3*1416, or where a great degree of accuracy is needed by 
3*1415926. 

POLYGONS, &0. 




Fig. 54. 



93. — TJjpon a given Ime to construct an equirlateral triangie. 
Let a 5, {Fig. 54,) be the given line. Upon a and 5, with a h 
for radius, describe arcs, intersecting at c/ join a and c, also c 
and 5 / then a ch will be the triangle required. 

94. — To describe an equi-lateral rectangle^ or sgua/re. Let 
a 5, {Fig, 55,) be the length of a side of the proposed square. 
Upon a and 5, with a h for radius, describe the arcs a d and 
he; bisect the arc, a e, in f ; upon ^, with e f for radius, de- 

5 



34 



AMERICAN HOTJSE-CABI'EIJTEB. 
r, a 




Fig. 55. 



scribe the arc, cfdj join a and c, c and d^ d and J / tlien a c 
d h will be the square required. 




95. — Within a given circle^ to inscribe cm equirlateral tri- 
angle^ hexagon or dodecagon. Let a h c d, {Fig. 56,) be the 
given circle. Draw the diameter, h d ; upon J, with the 
radius of the given circle, describe the arc, a e c j join a and 
c, also a and d^ and c and d — and the triangle is completed. 
For the hexagon: from a^ also from c, through ^, draw the 
lines, af and eg; join a and 5, h and c, c and /", &c., and the 
hexagon is completed. The dodecagon may be formed by 
bisecting the sides of the hexagon. 

Each side of a regular hexagon is exactly equal to the 
radius of the circle that circumscribes the figure. For the 
radius is equal to a chord of an arc of 60 degrees ; and, as 
every circle is supposed to be divided into 360 degrees, there 
is just 6 times 60, or 6 arcs of 60 degrees, in the whole circum- 
ference. A line drawn from each angle of the hexagon to the 
centre, (as in the figure,) divides it into six equal, equi-lateral 
triangles. 

96. — Within a square to inscrihe an octagon. Let ah c d^ 



PEACTICAL GEOMETET. 



35 




(Fig. 57,) be the given square. Draw the diagonals, a d and 
ho; upon a, 5, c and d^ with a e for radius, describe arcs cut- 
ting the sides of the square at 1, 2, 3, 4, 5, 6, 7 and 8 ; join 1 
and 2, 3 and 4, 5 and 6, &c., and the figure is completed. 

In order to eight-square a hand-rail, or any piece that is to 
be afterwards rounded, draw the diagonals, a d and h <?, upon 
the end of it, after it has been squared-up. Set a gauge to the 
distance, a e^ and run it upon the whole length of the stufi", 
from each corner both ways. This will show how much is to 
be chamfered off, in order to make the piece octagonal. {Art 
159.) 




Fig. 59 



97. — Within a given ci/rcle to inscribe any regular jpolygon. 
Let « J c 2, [Fig. 58, 59 and 60,) be given circles. Draw the 
diameter, ac; upon this, erect an equi-lateral triangle, a e c^ 
according to Art. 93 ; divide a c into as many equal parts as 
the polygon is to have sides, as at 1, 2, 3, 4, &c. ; from e. 



36 



AMERICAN HOTJSE-CARPENTEE. 



throiigli each even number, as 2, 4, 6, &c., draw lines cutting 

the circle in the points, 2, 4, &c. ; from these points and at 

right angles to a c, draw lines to the opposite part of the circle ; 

this will give the remaining points for the polygon, as h,f, &c. 

In forming a hexagon, the sides of the triangle erected upon 
a G, (as at Mg. 59,) mark the points h and /. This method of 
locating the angles of a polygon is an approximation suffi- 
ciently near for many purposes ; it is based upon the like prin- 
ciple with the method of obtaining a right hue nearly equal to 
a circle. {Art. 92.) The method shown at Art. 98 is accurate. 




Fig- 61. Fig. 62. Fig. 63. 

98. — Tlj^on a given line to describe am,y regular ^polygon. 

Let a 5, {Fig. 61, 62 and 63,) be given lines, equal to a side of 

the required figure. From 5, draw h c, at right angles to ah ; 

upon a and h, with a h for radius, describe the arcs, a c d and 

f eh I divide a c into as many equal parts as the polygon is to 

have sides, and extend those divisions from c towards d ; from 

the second point of division counting from c towards a, as 3, 

{Fig. 61,) 4, {Fig. 62,) and 5, {Fig. 63,) di-aw a line to h; take 

the distance from said point of division to a^ and set it from h 

to e^ join e and a j upon the intersection, o, with the radius, 

«, describe the circle a f d h j then radiating lines, drawn 

from h through the even numbers on the arc, a d^ will cut the 

circle at the several angles of the required figure. 

In the hexagon, {Fig. 62,) the divisions on the arc, a d, are not 
necessary ; for the point, c>, is at the intersection of the arcs, a d 
and fh, the points,/" and d, are determined by the intersection of 
those arcs with the circle, and the points above, g and A, can be 
found by drawing lines from a and h, through the centre, o. In 
polygons of a greater number of sides than the hexagon, the in- 
tersection, <?, comes above the arcs ; in such case, therefore, the 



PEACTICAL GEOMETRY. 37 

lines^ a e and h 5, {Fig. 63,) have to be extended before they will 
intersect. This method of describing polygons is fonnded on 
correct principles, and is therefore accurate. Li the circle equal 
arcs subtend equal angles, {Arts. 86 and 162.) Although this 
method is accurate, yet polygons may be described as accu- 
rately and more simply in the following manner. It will be 
observed that much of the process in this method is for the pur 
pose of ascertaining the centre of a circle that will circumscribe 
the proposed polygon. By reference to the Table of Polygons 
in the Appendix it will be seen how this centre may be obtained 
arithmetically. This is the Hule. — Multiply the given side by 
the tabular radius for polygons of a like number of sides with 
the proposed figure, and the product will be the radius of the 
required circumscribing circle. Divide this circle into as many 
equal parts as the polygon is to have sides, connect the points of 
division by straight lines, and the figure is complete. For exam- 
ple : It is desired to describe a polygon of Y sides, and 20 inches 
a side. The tabular radius is 1*1523824. This multiplied by 
20, the product, 23*04:764:8 is the required radius in inches. The 
Rules for the Reduction of Decimals, also in the Appendix, 
show how to change decimals to the fractions of a foot or an 
inch. From this, 23*04764:8 is equal to 23tV inches nearly. It 
is not needed to take all the decimals in the table, three or four of 
them will give a result snfiiciently near for all ordinary practice. 




Fig. 64. 

99. — To consi/ruct a triangle whose sides shall he severally 
equal to three given lines. Let «, h and <?, {Fig. 64,) be the given 
lines. Draw the line, d e^ and make it equal to c ; upon e^ with 
h for radius, describe an arc at f ; upon d^ with a for radius, 
describe an arc intersecting the other at // join d and /, also 
f and e ; then df e will be the triangle required. 




rig. 65. Fig. 66. 



38 



PRACTICAL GEOMETRY. 



100. — To construct a figure equal to a given, right-lined 

figure. Let abed, {Fig. 65,) be the given figure. Make ef, 

{Fig. 66,) equal to c d ; upon /, with d a for radius, describe an 

arc at g; upon e, with c a for radius, describe an arc intersecting 

the other at g ; join g and e ; upon / and g, with d h and a b 

for radius, describe arcs intersecting at h ; join g and A, also h 

and / ; then Fig. 66 will every way equal Fig. 65. 

So, right-lined figures of any number of sides may be copied, 
oy first dividing them into triangles, and then proceeding as 
above. The shape of the floor of any room, or of any piece of 
land, &c., may be accurately laid out by this problem, at a scale 
upon paper ; and the contents in square feet be ascertained by 
the next. 




101. — To make a parallelogram equal to a given triangle. 

Let a b c, {Fig. 67,) be the given triangle. From a, draw a d, 

at right angles to be; bisect a d in e ; through e, draw/^, 

parallel to 6 c ; from 6 and c, draw b f and c g, parallel to d e ; 

then bfgc will be a parallelogram containing a surface exactly 

equal to that of the triangle, a b c 

Unless the parallelogram is required to be a rectangle, the lines, 
b f and c g, need not be drawn parallel to d e. If a rhomboid is 
desired, they may be drawn at an oblique angle, provided they 
be parallel to one another. To ascertain the area of a triangle, 
multiply the base, b c, by half the perpendicular height, d a. In 
doing this, it matters not which side is taken for base. 



A ^ 

^y^ e 
^^ C 



d 
Fig. 68. 



AMERICAN HOUSE-CARPENTER. 



39 



i02.—A parallelogram being given, to construct anothel 
equal to it, and having a side equal to a given line. Let A 
{Fig. 68,) be the given parallelogram, and B the given line 
Produce the sides of the parallelogram, as at a, b, c and d ; make 
e d e:iual to B ; through d, draw c /, parallel to g b ; through 
e, draw the diagonal, c a ; from a, draw a /, parallel to e d 
then C will be equal to A. (See Art. 144.) 



A 


a ^\i 




B 



Fig 69. 

103. — To make a square equal to two or more given squares. 
Let A and B, {Fig. 69,) be two given squares. Place them sc 
ds to form a right angle, as at a ; join b and c ; then the square, 
C, formed upon the line, b c, will be equal in extent to the squares, 
A and B, added together. Again : if a 6, {Fig. 70,) be equal to 




the side of a given square, c a, placed at right angle'J to a b, he the 
side of another given square, and c d, placed at right angles to 



40 PRACTICAL GEOMETRY. 

c b, be the side of a third given, square ; then the square, A, 
formed upon the Une, d b, will be equal to the three given 
squares. (See Art. 157.) 

The usefulness and importance of this problem are proverbial. 
To ascertain the length of braces and of rafters in framing, the 
length of stair-strings, &c., are some of the purposes to which it 
may be applied in carpentry. (See note to Art. 74., b.) If the 
length of any two sides of a right-angled triangle is known, that 
of the third can be ascertained. Because the square of the 
hypothenuse is equal to the united squares of the two sides that 
contain the right angle. 

(1.) — The two sides containing the right angle being known, 
to find the hypothenuse. Rule. — Square each given side, add 
the squares together, and from the product extract the square- 
root : this will be the answer. For instance, suppose it were 
required to find the length of a rafter for a house, 34 feet wide, — 
the ridge of the roof to be 9 feet high, above the level of the 
v/all-plates. Then 17 feet, half of the span, is one, and 9 feet, 
the height, is the other of the sides that contain the right angle 
Proceed as directed by the rule : 

17 9 

17 9 



119 81 = square of 9. 

17 289 = square of 17. 



289 = square of 17. 370 Product. 

1 ) 370 ( 19-235 -f = square-root of 370 ; equal 19 feet, 2^ in. 
1 1 nearly : which would be the required 

— length of the rafter. 

29 ) 270 
9 261 



382) ••900 
2 764 



3843 ) 13600 
3 11529 



38465)* 207100 (By reference to the table of square-roots 
192325 in the Appendix, the root of almost any 

number may be found ready calenlntfid ; 

also, to change the decimals of a foot to inches and parts, see 
Eules for the Keduction of Decimals in the Appendix.) 



AMERICAN HOUSE-CARPENTER. 41 

Agair. : suppose it be required, in a frame building, to nnd the 
lenp-th of a brace, having a run of three feet each way from the 
point of the right angle. The length of the sides containing the 
right angle will be each 3 feet : then, as before — 

3 
3 

9 = square of one side. 
3 times 3 = 9 = square of the other side. 

1 8 Product : the square-root of which is 4*2426 + ft., 
or 4 feet, 2 inches and ^ths. full. 

(2.) — The hypothenuse and one side being known, to find the 
other side. Rule. — Subtract the square of the given side from 
the square of the hypothenuse, and the square-root of the product 
will be the answer. Suppose it were required to ascertain the 
greatest perpendicular height a roof of a given span may have, 
when pieces of timber of a given length are to be used as rafters. 
Let the span be 20 feet, and the rafters of 3x4 hemlock joist. 
These come about 13 feet long. The known hypothenuse, 
then, is 13 feet, and the known side, 10 feet — that being half the 
span of the building. 

13 
13 



39 
13 



169 = square of hypothenuse, 
10 times 10 = 100 = square of the given side. 

69 Product : the square-root of which is 8 
•3066 + feet, or 8 feet, 3 inches and »-ths. full. This will be 
the greatest perpendicular height, as required. Again : suppose 
that in a story of 8 feet, from floor to floor, a step-ladder is re- 
quired, the strings of which are to be of plank, 12 feet long ; and 
it is desirable to know the greatest run such a length of string 
will afford. In this case, the two given sides are — hypothenuse 
12, perpendicular 8 feet. 

12 times 12 = 144 = square of hypothenuse. 
8 times 8 = 64 = square of perpendicular. 

80 Product : the square-root of which is 8-9442 -f 
feet, or 8 feet, "^ 1 inches and ^^ihs. — the answer, as required. 



^1 



PRACTICAL GEOMETRY. 



Many other cases might be adduced to show the utility of this 
problem. A practical and ready method of ascertaining the 
length of braces, rafters, (fee, when not of a great length, is to 
apply a rule across the carpenters'-square. Suppose, for the 
length of a rafter, the base be 12 feet and the height 7. Apply 
the rule diagonally on the square, so that it touches 12 inches 
from the corner on one side, and 7 inches from the corner on the 
other. The number of inches on the rule, which are intercepted 
by the sides of the square, 13 f nearly, will be the length of the 
rafter in feet ; viz, 13 feet and Jths of a foot. If the dimensions 
are large, as 30 feet and 20, take the half of each on the sides of 
the square, viz, 15 and 10 inches ; then the length in inches 
across, will be one-half the number of feet the rafter is long. 
This method is just as accurate as the preceding ; but when 
the length of a very long rafter is sought, it requires great care 
and precision to ascertain the fractions. For the least variation 
on the square, or in the length taken on the rule, would make 
perhaps several inches difference in the length of the rafter. 
For shorter dimensions, however, the result will be true enough. 




Fi-. 71. 



104. — To make a circle equal to two given circles. Let A 
and B^ {Fig. 71,) be the given circles. In the right-angled tri- 
angle, a b c, make a b equal to the diameter of the circle. B, and 
b equal to the diameter of the circle, A ; then the hypothenuse. 




Fig. 78. 



AMERICAN HOUSE-CARPENTER. 40 

a c, will be the diameter of a circle, C, which will be equal in 

area to the two circles, A and B^ added together. 

Any polygonal figure, as A, [Fig. 72,) formed on the hypo- 
thenuse of a right-angled triangle, will be equal to two similar 
figures,* as B and C, formed on the two legs of the triangle. 




Fig. 73 

105. — To construct a square equal to a given rectangle. 
Let A, {Fig. 73,) be the given rectangle. Extend the side, a 6, 
and make b c equal tob e ; bisect a c in/, and upon/, with the 
radius, / a, describe the semi-circle, age; extend e 6, till it 
cuts the curve in g ; then a square, b g h d^ formed on the line, 
h fi-, will be equal in area to the rectangle, A, 



e 

l 

A 



^ g 

Fig. 74. 



105, a. — Another method. Let A^ {F^g' 7'4,) be the given 
rectangle. Extend the side, a b, and make a d equal to a c ; 

* Similar figures are such as have their several angle* respectively equal, and their 
rfiiljs respectively proportionate. 



4A PRACTICAL GEOMETRY. 

biseiit a d in e ; upon e, with the radius, e a, describe the seim 
circle, afd; extend^ b till it cuts the curve in/; join a and 
f; then the square, B, formed on the line, a/, will be equal in 
area to the rectangle, A. (See Art. 156 and 157.) 

106. — To form a square equal to a given triangle. Let a b, 
{Mg. 73,) equal the base of the given triangle, and b e equal 
half its perpendicular height, (see Fig. 67 ;) then proceed a? 
directed at Art. 105. 




Fie. 75. 



107. — Two right lines being given, to find a third propor- 
tional thereto. Let A and B, [Fig. 75,) be the given lines. 
Make a b equal to A ; from a, draw a c, at any angle with ab ; 
make a c and a d each equal to B ; join c and b ; from d, draw 
d e, parallel to c b ; then a e will be the third proportional re- 
quired. That is, a e bears the same proportion to B, as B does 
to A. 




108. — Three right lines being given, to find a fourth pro 
vortional thereto. Let A, B and C, [Fig. 7^,) be the given 
lines. Make a b equal to A ; from a, draw a c, at any angle 
with a b; make a c equal to B, and a e equal to C ; join c and 
b ; from e, draw ef parallel to c b ; then af will be the fourth 
proportional required. That is, a f bears the same proportion 
to C, as B does to A, 



AMERICAN HOUSE-CARPENTER. 



45 



To apply this problem, suppose the two axes of a given ellipsis 
and the longer axis of a proposed ellipsis are given. Then, by 
this problem, the length of the shorter axis to the proposed ellip- 
sis, can be found ; so that it will bear the same proportion to the 
longer axis, as the shorter of the given ellipsis does to its longer. 
(See also, Art. 126.) 



A 

P 
















\^ 


^\ 


a 


1 


2 
Fig 


3 

77. 


4 5 \ 



109. — A line with certain divisions being given, to divide 

another, longer or shorter, given line in the same proportion. 

Let A, {Fig. 77,) be the line to be divided, and B the line with 

its divisions. Make a h equal to B, with all its divisions, as at 

1,2, 3, &c. ; from a, draw a c, at any angle with a b ; make a c 

equal to A ; join c and b ; from the points, 1, 2, 3, (fee, draw 

lines, parallel to c b ; then these will divide the line, a c, in the 

same proportion as B is divided — as was required. 

This problem will be found useful in proportioning the mem- 
bers of a proposed cornice, in the same proportion as those of a 
given cornice of another size. (See Art. 253 and 254.) So of 
a pilaster, architrave, &c. 




Fig. 78. 



110. — Between two given right lines^ to find a mean pi o- 
portional. Let A and Bj {Fig. 78,) be the given lines. On 
tlie line, a c, make a b equal to A, and b c equal to B ; bisect a 
c in e ; upon e, with e a for radius, describe the semi circle, a d 



46 



AMERICAN HOTJSE-CARPENTEE. 



c ; at h, erect h d, at riglit angles to ac; then h d will be tlio 
mean proportional between A and B. That is, <3^ 5 is to 5 (^ as 
h d istoh c. This is nsaally stated thus— «^ h :1? b::hd ih c, 
and since the product of the means equals the product of the 
extremes, therefore, ahxh G= Td\ Tliis is shown geometri- 
cally at Art. 105. 

CONIC SECTIONS. 

111. — Jf a cone, standing upon a base that is at right angles 
with its axis, be cut by a plane, perpendicular to its base and 
passing through its axis, the section will be an isosceles triangle ; 




Fig. 79. 



\ais ahc, Fig. 79 ;) and the base will be a semi-circle. If a cone 
be cut by a plane in the direction, e /", the section will be an 
elUjjsis / if in the direction, m Z, the section will be 2i> parabola ; 
and if in the direction, r c, an hyperbola. (See Art. 56 to 60.) If 
the cutting planes be at right angles with the plane, ahc^ then — 
112. — To find the axes of the ellipsis,, bisect ef {Fig. Y9,) in 
g I through ^, draw h i, parallel to ah ; bisect h i inj; upon 
J, with j h for radius, describe the semi-circle, hh i / from g, 
draw g h^ at right angles to h i ^ then twice g h will be the 
conjugate axis, and ^ythe transverse. 



AMERICAN HOUSE-CARPENTER. 



47 



113. — To find the axis and base of the parabola. Let m U 
[Fig. 79j) parallel to a c, be the direction of the cutting plane. 
From m, draw m d, at right angles to a 6 ; then I m will be the 
axis and height, and m d an ordinate and half the base ; as at 
Fig. 92, 93. 

114. — To find the height^ base and transverse axis of an 
hyperbola. Let o r, [Fig. 79,) be the direction of the cutting 
plane. Extend o r and a c till they meet at n ; from o, draw 
;?, at right angles to a b; then r o will be the height, n r the 
transverse axis, and o p half the base ; as at Fig. 94. 




Fig. 80. 



115. — The axes being given, to find the foci, and to describe 

an ellipsis with a string. Let a b, [Fig. 80,) and c d, be the 

given axes. Upon c, with a e orb e for radius, describe the arc, 

ff; then /and/, the points at which the arc cuts the transverse 

axis, will be the foci. At/ and /place two pins, and another at c ; 

tie a string about the three pins, so as to form the triangle, ffc ; 

remove the pin from c, and place a pencil in its stead ; keeping the 

string taut, move the pencil in the direction, eg a; it will then 

describe the required ellipsis. The lines, fg and g f show the 

position of the string when the pencil arrives at g. 

This method, when performed correctly, is perfectly accurate ; 
but the string is liable to stretch, and is, therefore, not so good to 
use as the trammel. In making an ellipse by a string or twine, 
that kind should be used which has the least tendency to elasticity. 
For this reason, a cotton cord, such as chalk-lines are commonly 
made of, is not proper for the purpose : a linen, or flaxen cord is 
mu3h better. 



48 PRACTICAL GEOMETRY. 




Fis. SI 



116. — The axes being given,, to describe an ellipsis with a 
trammel. Let a b and c c?, [Fig. 81,) be the given axes. Place 
the trammel so that a line passing through the centre of the 
grooves, would coincide with the axes ; make the distance from 
the pencil, e, to the nut,/, equal to half c d ; also, from the pen- 
cil, e, to the nut, g^ equal to half ab ; letting the pins under the 
nuts slide in the grooves, move the trammel, e g^ in the direction, 
c b d ; then the pencil at e will describe the required ellipse. 

A trammel may be constructed thus : take two straight strips ot 
board, and make a groove on their face, in the centre of their 
width ; join them together, in the middle of their length, at right 
angles to one another ; as is seen at Fig. 81. A rod is then to be 
prepared, having two moveable nuts made of wood, with a mor- 
tice through them of the size of the rod, and pins under them 
large enough to fill the grooves. Make a hole at one end of the 
rod, in which to place a pencil. In the absence of a regular tram- 
mel, a temporary one may be made, which, for any short job, 
will answer every purpose. Fasten two straight-edges at right 
angles to one another. Lay them so as to coincide with the axes 
of the proposed ellipse, having the angular point at the centre. 
Then, in a rod having a hole for the pencil at one end, place two 
brad-awls at the distances described at Art. 116. While the 
pencil is moved in the direction of the curve, keep the brad-awls 
hard against the straight-edges, as directed for using the tram- 
mel-rod, and one-quarter of the ellipse will be drawn. Then, 
by shifting the straight-edges, the other three quarters in succes- 
sion may be drawn. If the required ellipse be not too large, a 
carpenters'-square may be made use of, in place of the straight- 
edges. 

An improved method of constructing the trammel, is as fol 
lows : make the sides of the grooves bevilling from the face of 
the stuff, or dove-tailing instead of square. Prepare two slips ot 
wood, each about two inches long, which shall be of a shape to 
just fill the groove when slipped in at the end. These, instead of 



AMERICAN HOUSE-CARPENTER. 



49 



pins, are to be attached one to each of the moveable nuts with 
a screw, loose enough for the nut to move freely about the screw 
as an axis. The advantage of this contrivance is, in preventing 
the nuts from slipping out of their places, during the operation 



of describing the curve. 




Fig. 82. 

117. — To describe an ellipsis hy ordinates. Let a h and c a, 
{Pig- 82,) be given axes. With c e or e d for radius, de- 
scribe the quadrant,/^ h ; divide f h, a e and e b, each into a 
like number of equal parts, as at 1 , 2 and 3 ; through these 
points, draw ordinates, parallel to c d and fg ; take the distance, 
1 ^, and place it at 1 Z, transfer 2 j to 2 m, and 3 A: to3 w; through 
the points, a, 7^, m, I and c, trace a curve, and the ellipsis will 
be completed. 

The greater the number of divisions on a e, &c., in this and 
the following problem, the more points in the curve can be found, 
and the more accurate the curve can be traced. If pins are 
placed in the points, w, m, /, &c., and a thin slip of wood bent 
around by them, the curve can be made quite correct. This 
method is mostly used in tracing face-moulds for stair hand- 
railing. 




118. — To desaHbe an ellipsis by intersection of lines. 

7 



Lo.% 



50 



PRACTICAL GEOMETRY. 



a b and c d, {Fig. 83,) be given axes. Through c, draw / ^, 
parallel Xc ah ; from a and 6, draw a f and h g, at right angles 
to ab ; divide f a, g b, a e and e 6, each into a like number of 
equal parts, as at 1, 2. 3 and o, o, o ; from 1, 2 and 3, draw lines 
to c ; through o, o and o, draw lines from d, intersecting those 
drawn to c ; then a curve, traced through the points, i, i, i, will 
be that of an ellipsis. 




Where neither trammel nor string is at hand, this, perhaps, is 
the most ready method of drawing an ellipsis. The divisions 
should be small, where accuracy is desirable. By this method, 
an ellipsis may be traced without the axes, provided that a diame- 
ter and its conjugate be given. Thus, a b and c d, {Fig. 84,) are 
conjugate diameters : f g is drav/n parallel to a b, instead of 
being at right angles to c d ; also,/ a and ^ b are drawn paraUel 
to c 5, instead of being at right angles to a b. 




119. — To describe an ellipsis by intersecting arcs. I^t a b 



AMERICAN HOUSE-CARPENTER. 



51 



ana c c?, [Fig. 85,) be given axes. Between one of the foci, / 
and/, and the centre, e, mark any number of points, at random, 
as 1, 2 and 3 ; upon /and/, with h 1 for radius, describe arcs at 
g^ g, g and g ; upon/ and/, with a 1 for radius, describe arcs inter- 
secting the others at g,g,g and g ; then these points of intersection 
will be in the curve of the eUipsis. The other points, h and i, are 
found in like manner, viz: h is found by taking 6 2 for one radius, 
and a 2 for the other ; i is found by taking b 3 for one radius, and 
a 3 for the other, always using the foci for centres. Then by 
tracing a curve through the points, c, g, A, i, b, &c., the ellipse 
will be completed. 

This problem is founded upon the same principle as that of the 
string. This is obvious, when we reflect that the length of the 
string is equal to the transverse axis, added to the distance between 
the foci. See Fig. 80; in which c/ equals a e, the half of the 
transverse axis. 




Fig. 86. 

120. — To describe a figure nearly in the shape of an ellip- 
sis , by a pair of compasses. Let a b and c d^ {Fig. 86,) be 
given axes. From c, draw c e, parallel to ab ; from a, draw a e, 
parallel io c d; join e and c?; bisect e a in/; join/ and c, inter- 
secting e dini; bisect i cin o ; from o, draw og, at right angles 
to i c, meeting c d extended to g ; join i and g, cutting the trans- 
verse axis in r ; make h j equal to h g, and h k equal to A r ^ 
from j. through r and k^ d'aw^* m and 7 n ; also, from g, through 
k,arawgl; upon ^ and 7, with ^ c for radius, describe the 



52 



PRACTICAL GEOMETRY 



arcs, i I and m n ; upon r an 1 A:, with r a for radius, describe 
the arcs, ?7i i and Z 7i , this will complete the figure. 

When the axes are proportioned to one another as 2 to 3, the 
extremities, c and c?, of the shortest axis, will be the centres for 
describing the arcs, i I and m n ; and the intersection of e d with 
the transverse axis, will be the centre for describing the arc, m z, 
(fee. As the elliptic curve is continually changing its course from 
that of a circle, a true ellipsis cannot be described with a pair ot 
compasses. The above, therefore, is only an approximation. 




121. — To draio an oval in the j^^oportion^ seven by nine. 
Let c d^ {Fig. 87,) be the given conjugate axis. Bisect c c? in o, 
and through o, draw a 6, at right angles to c d ; bisect c o in e , 
upon 0, with o e for radius, describe the circle, e f g h ; from e, 
through h and/, draw e j and e i ; also, from^, through li and/, 
draw g k and g I ; upon g^ with g c for radius, describe the arc, 
k I ; upon e, with e d for radius, describe the arc, j i ; upon h and 
/", with h k for radius, describe the arcs, j k and I i ; this will 
complete the figure. 

This is an approximation to an ellipsis ; and perhaps no 
method can be found, by which a well-shaped oval can be drawn 
with greater facility. By a little variation in the piocess, ovals 
of different proportions may be obtained. If quarter cf the trans- 
verse axis is taken for the radius of the circle, efg h, one will be 
drawn in the proportion, five by seven. 



AMERICAN HOUSE-CARPENTER. 



53 




122.— To draw a tangent to an ellipsis. Let a 6 c d, {Fig. 
88,) be the given ellipsis, and d the point of contact. Find the 
foci, {Art. 115,)/ and/, and from them, through d, draw/e and 
f d; bisect the angle, {Art. 77,) e d o, with the line, s r : then 
s T will be the tangent required. 




c Fig 89 



123. — An ellipsis with a tangent given, to detect the point 
of contact. Ijetagbf, {Fig. 89,) be the given ellipsis and tan- 
gent. Through the centre, e, draw a b, parallel to the tangent ; 
any where between e and/, draw c d, parallel to ab ; bisect cdin 
; through o and e, draw/ ^ ; then g will be the point of con- 
tact required. 

124. — A diameter of an ellipsis given, to find its conjugate. 
Let a b, {Fig. 89,) be the given diameter. Find the ]me,fg, by 
the last problem ; then/^ will be th? diameter required. 



PRACTICAL GEOMETRY. 




125. — Any diameter and its conjugate being given, to as- 
certain the two axes, and thence to describe the ellipsis, ijet 
a b and c d, {Fig. 90,) be the given diameters, conjugate to one 
another. Through c, draw e /, parallel to a b ; from c, draw c 
g, at right angles to ef; make c g equal to a h ot h b ; join g 
and h ; upon g, with g c for radius, describe the arc, i k c j ; 
upon h, with the same radius, describe the arc, In; through the 
intersections, I and n, draw n o, cutting the tangent, ef, in o ; 
upon 0, with o ^for radius, describe the semi-circle, eigf; join 
€ and^, also g and/, cutting the arc, i c j, in k and t ; from e, 
through ^, draw e m, also from/, through A, draw/p ; from k 
and ^, draw ^ r and ^ 5, parallel to g h, cutting e ni in r, and/p 
in s ; make h m equal to A r, and A p equal to h s ; tlien r m 
and 5 p will be the axes required, by which the ellipsis may be 
drawn in the usual way. 

126. — To describe an ellipsis, whose axes shall be propor- 
tionate to the axes of a larger o-^ smaller given one. Let a 
cbd, {Fig- 91,) be the given ellipMS and axes, and i j the trans- 
verse axis of a proposed smaller one. Join a and c ; trom i, 
Iraw i e, parallel to a c ; make o / equal to o e ; then ef will be 



AMERICAN HOUSE-CARPENTER* 



55 




Fig. 91. 



the conjugate axis required, and will bear the same proportion to 
' j, asc d does to a b. (See Art. 108.) 



1 2 3 i 3 2 1 




127. — To describe a parabola by intersection of lines. Let 
m Ij {Fig. 92j) be the axis and height, (see Fig. 79,) and d d, a. 
double ordinate and base of the proposed parabola. Through L 
draw a a, parallel to d d ; through d and d, draw d a and d a, 
parallel to ni I ; divide a d and d m, each into a like number oi 
equal parts ; from each point of division in d m, draw the lines, 
1 1, 2 2, (fee, parallel to ml; from each point of division in d 
a, draw lines to I ; then a curve traced through the points ot 
intersection, o, o and o, will be that of a parabola. 

127, a. — Another method. Let m, I, {Fig. 93,) be the axis and 
height, and d d the base. Extend m I, and make I a equal to m> 
I ; join a and rf, and a and d ; divide a d and a d. each into a 
liKe number of equal parts, as at 1, 2, 3, &c. ; join 1 and 1, 2 and 
2, (fee, and the parabola will be completed 



56 



PRACTICAL GEOMETRY. 




Fig. 93. 




128. — To describe an hyperbola by intersection of lines. 
Let r 0, {Fig. 94,) be the height, p p the base, and n r the trans- 
verse axis. (See Fig, 79.) Through r, draw a a, parallel to p 
p : from p, draw a _p, parallel to r o ; divide a p and p o, each 
into a like number of equal parts ; from each of the points of di- 
visions in the base, draw lines to n ; from each of the points of 
division in a p, draw lines to r ; then a curve traced through the 
points of intersection, o, o, &c., will be that of an hyperbola. 

The parabola and hyperbola afford handsome curves for various 
mo il dings. 



DEAIONSTRATIOWS. 



129. — To impress more deeply upon the mind of the learnei 
some of the more important of the preceding problems, and to 
indulge a very common and praiseworthy curiosity to discover 
the cause of things, are some of the reasons why the following 
exercises are introduced. In all reasoning, definitions are ne- 
cessary ; in order to insure, in the minds of the proponent and 
respondent, identity of ideas. A corollary is an inference deduced 
from a previous course of reasoning. An axiom is a proposition 
evident at first sight. In the following demonstrations, there are 
many axioms taken for granted ; (such as, things equal to the 
same thing are equal to one another, &c. ;) these it was thought 
not necessary to introduce in form. 



b 

Fig. 95. 



130. — Definition. If a straight line, as a h, {Fig. 95,) stand 

upon another straight line, as c d, so that the two angles made at 

8 



68 PRACTICAL GEOMETRY. 

the point, 6, are equal — a b do a b d, (see note to Art. 27,) then 
each of the two angles is called a right angle. 

131. — Definition. The circumference of every circle is sup- 
posed to be divided into 360 equal parts, called degrees ; hence 
a semi-circle contains 180 degrees, a quadrant 90, (fee. 




Fig. 96. 

132. — Definition. The measure of an angle is the number of 
degrees contained between its two sides, using the angular point 
as a centre upon which to describe the arc. Thus the arc, c e, 
{Fig. 96,) is the measure of the angle, c b e ; e a, of the angle, 
e b a ; and a d, of the angle, ab d. 

133. — Corollary. As the two angles at 6, {Fig. 95,) are right 
angles, and as the semi-circle, cad, contains 180 degrees, {Art. 
131,) the measure of two right angles, therefore, is 180 degrees ; 
of one right angle, 90 degrees ; of half a right angle, 45 ; of 
one-third of a right angle, 30, ifec. 

134. — Defiadtion. In measuring an angle, {Art. 132,) no re- 
gard is to be kad to the length of its sides, but only to the degree 
of their inclination. Hence equal angles are such as have the 
same degree of inclination, without regard to the length of their 
sides. 




135. — Axiom. If two straight lines, parallel to one another, 



AMERICAN HOUSE-CARPENTER. 



59 



3ls a b and c d, {Fig. 97,) stand upon another straight line, as ef, 
the angles, a bf and c d f^ are equal ; and the angle, a 6 e, is 
equal to the angle, c d e. 

136. — Definition, If a straight line, as a 6, [Fig. 96,) stand 
obliquely upon another straight line, as c d, then one of the an- 
gles, as a b c, is called an obtuse angle, and the other, as ab d, 
an acute angle. 

137. — Axiom. The two angles, a b d and a b c, {Fig. 96,) are 
together equal to two right angles, {Art. 130, 133 ;) also, the 
three angles, a b d,e b a and c 6 e, are together equal to two right 
angles. 

138. — Corollary. Hence all the angles that can be made upon 
one side of a line, meeting in a point in that line, are together 
equal to two right angles. 

139. — Corollary. Hence all the angles that can be made on 
both sides of a line, at a point in that line, or all the angles that 
can be made about a point, are together equal to four right angles. 





Fig 98. 



140. — Proposition. If to each of two equal angles a third 
angle be added, their sums will be equal. Let ab c and d efi 
{Fig. 98,) be equal angles, and the angle, i j k, the one to be 
added. Make the angles, gb a and hed, each equal to the given 
angle, ijk; then the angle, g b c, will be equal to the angle, h e 
f ; for, \i ab c and d e/be angles of 90 degrees, and i j k, 30, 
then the angles, g b c and h ef, will te each equal to 90 and 
30 added, viz : 120 degrees. 



60 



PRACTICAL GEOMETRY. 
a d 





Fig. 99. 

141. — Proposition. Triangles that have two of then* sides 
and the angle contained between them respectively equal, have 
also their third sides and the two remainmg angles equal ; and 
consequently one ti'iangle will every way equal the other. Let a 
b c, {Fig-. 99,) and d efhe two given triangles, havmg the angle 
at a equal to the angle at d, the side, a b, equal to the side, d e, 
and the side, a c, equal to the side, df; then the third side of 
one, b c, is equal to the third side of the other, ef; the angle at b 
is equal to the angle at e, and the angle at c is equal to the angle 
at/. For, if one triangle be applied to the other, the three points, 
6, a, c, coinciding with the three points, e, d, f, the line, b c, must 
coincide with the Ime, e f; the angle at b with the angle at e ; 
the angle at c with the angle at/ ; and the triangle, 6 a c, be every 
way eoual to the triangle, e df. 




142. — Proposition. The two angles at the base of an isoceles 
triangle are equal. Let a 6 c, {Fig. 100,) be an isoceles triangle, 
of which the sides, a b and a c, are equal. Bisect the angle, {Art 



AMERICAN HOUSE-CARPENTER. 



ei 



77,) 6 a c, by the line, a d. Then the line, h a, being equal to 
the line, a c ; the Mne, a d, of the triangle, A^ being equal to the 
line, a rf, of the triangle, B^ being conmion to each ; the angle, b 
a dj being equal to the angle, d a c ; the line, b d, must, accord- 
ing to Art. 141, be equal to the line, dc; and the angle at b mus* 
be equal to the angle at c. 



A 


^^ 




/ 


^v^ 


V 




B 


i 


/ 


B 


</ 


c 


i 


d 




c 






Fis 


101. 







143. — Proposition. A diagonal crossing a parallelogram di- 
vides it into two equal triangles. Let a b c d^ [Fig. 101,) be a 
given parallelogram, and 6 c, a line crossing it diagonally. Then, 
as a c is equal to b d, and a b to c d, the angle at a to the angle 
at d, the triangle. A, must, according to Art. 141, be equal to the 
triangle, B. 



A 


^^y 
^y^^ 




B 




Fig. 102. 

144. — Proposition. Let abed, {Fig. 102,) be a given pa- 
rallelogram, and 6 c a diagonal. At any distance between a b and 
c d, draw e /, parallel to ab; through the point, g, the intersection 
of the lines, b c and e/, draw h i, parallel to b d. In every paral- 
lelogram thus divided, the parallelogram, A, is equal to the paral- 
lelogram, B. According to Art. 143, the triangle, a 6 c, is 
equal to the triangle, bed; the triangle, Cj to the triangle, D ; 
and E to F ; this being the case, take D and F from the triangle, 
h c d, and C and E from the triangle, ab c, and what remains 



62 



PRACTICAL GEOMETRY. 



in one must be equal to what remains in the other; therefore, the 
parallelogram, A, is equal to the parallelogram, B. 




Fig. 101 



145. — Proposition. Parallelograms standing upon the same 
base and between the same parallels, are equal. Let abed and 
efcd, {Fig. 103,) be given parallelograms, standing upon the 
same base, c d, and between the same parallels, a f and c d. 
Then, ab and e/ being equal to c d, are equal to one another; 
b e being added to both a b and ef, a e equals bf; the line, ac, 
being equal to b d, and a e to bf, and the angle, c a e, being 
equal, {Art. 135,) to the angle, d bf, the triangle, a e c, must be 
equal, {Art. 141,) to the triangle, bfd; these two triangles being 
equal, take the same amount, the triangle, beg, from each, and 
what remains in one, a b g c, must be equal to what remains in 
the other, efdg; these two quadrangles being equal, add the 
same amount, the triangle, c ^ c^, to each, and they must still be 
equal ; therefore, the parallelogram, ab c d,\s equal to the paral- 
lelogram, ef c d. 

146. — Corollary. Hence, if a parallelogram and triangle stand 
upon the same base and between the same parallels, the parallelo- 
gram will be equal to double the triangle. Thus, the paral- 
lelogram, a d, {Fig. 103,) is double, {Art. 143,) the triangle, 
c e d. 

147. — Proposition. Let abed, {Fig. 104,) be a given quad- 
rangle with the diagonal, a d. From b, draw b e, parallel toa d; 
extend cdto e ; join a and e ; then the triangle, a ee, will be equal 
in area to the quadrangle, abed. Since the triangles, adb and 
a d e, stand uj-on the same base, a d, and between the same paral- 



AMERICAN HOUSE-CARPENTER. 63 




lels, a d and h e, they are therefore equal, [Art. 145, 146 ;) and 
since the triangle, C, is common to both, the remaining triangles, A 
and -B, are therefore equal ; then B being equal to A^ the triangle, 
a e c, is equal to the quadrangle, abed. » 




148. — Proposition. If two straight lines cut each other, as 
a h and c c?, [Fig. 105,) the vertical, or opposite angles, A and 
C, are equal. Thus, a e, standing upon c c?, forms the angles, 
B and O, which together amount, [Art. 137,) to two right angles ; 
in the same manner, the angles, A and 5, form two right angles ; 
since the angles, A and jB, are equal to B and C, take the same 
amount, the angle, 5, from each pair, and what remains of one 
pair is equal to what remains of the other ; therefore, the an- 
gle, A, is equal to the angle, C. The same can be proved of 
the opposite angles, B and D. 

149. — Proposition. The three angles of any triangle are 
equal to two right angles. Let a 6 c, {Fig. 106,) be a given tri- 
angle, with its sides extended to/, e, and c?, and the line, cg^ 



64 PRACTICAL GEOMETRY. 




drawn parallel tob e. As g c is parallel to e 6, the angle, g c d^ 
is, equal, {Art. 135,) to the angle, e hd ; as the lines, /c and h e, 
cut one another at a, the opposite angles, f a e and b a c, are 
equal, {Art. 148 ;) as the angle, /a e, is equal, {Art. 135,) to the 
angle, acg, the angle, a c ^, is equal to the angle, b a c ; there- 
fore, the three angles meeting at c, are equal to the three angles 
of the triangle, a b c ; and since the three angles at c are equal, 
{Art. 137,) to two right angles, the three angles of the triangle, a 
b c, must likewise be equal to two right angles. Any triangle 
can be subjected to the same proof. 

150. — Corollary. Hence, if one angle of a triangle be a right 
angle, the other two angles amount to just one right angle. 

151. — Corollary. If one angle of a triangle be a right angle, 
and the two remaining angles are equal to one another, these are 
each equal to half a right angle. 

152. — Corollary. If any two angles of a triangle amount to 
a right angle, the remaining angle is a right angle. 

153. — Corollary. If any two angles of a triangle are together 
equal to the remaining angle, that remaining angle is a right 
angle. 

154. — Corollary. If any two angles of a triangle are each 
equal to two-thirds of a right angle, the remaining angle is also 
equal to two-thirds of a right angle. 

155. — Corollary. Hence, the angles of an equi-lateral trian- 
gle, are each equal to two- thirds of a right angle. 



AMERICAN HOUSE-CARPENTER. 
b 



65 




Fig. 107. 



156. — Proposition. If from the extremities of the diameter of 
a semi-circle, two straight lines be drawn to any point in the cir- 
cumference, the angle formed by them at that point wiVi be a 
right angle. Let ah c, {Fig. 107,) be a given semi-circle , and 
a b and h c, lines drawn from the extremities of the diameter, a 
c, to the given point, h ; the angle formed at that point by these 
lines, is a right angle. Join the point, 6, and the centre, d ; the 
lines, d a, d b and d c, being radii of the same circle, are equal ; 
the angle at a is therefore equal, {Art. 142,) to the angle, ab d, 
also, the angle at c is, for the same reason, equal to the angie, d b 
c ; the angle, a b c, being equal to the angles at a and c taken 
together, must therefore, {Art. 153,) be a right angle. 




Fiir. 



J 57. — Proposition. The square of the hypothenuse of a 

right-angled triangle, is equal to the squares of the two remaining 

sides. Let a b c, {Fig-. 108,) be a given right-angled triangle, 

having a square formed on each of its sides : then, the square, b e, is^ 

equal to the squares, h c and g 6, taken together. This can be 

9 



GQ PRACTICAL GEOMETRY. 

proved by showing that the paiallelogram, b Z, is equal to the square, 
o-b ; and that the parallelogranij c I, is equal to the square, h c. The 
angle, c 6 o?, is a right angle, and the angle, a 6 /, is a right angle ; 
add to each of these the angle, ab c ; then the angle,/ b c, will evi- 
dently be equal, {Art. 140,) to the angle, ab d ; the triangle,/^ c, 
and the square, g b, being both upon the same base,/ 6, and between 
the same parallels, / b and^ c, the square, g b, is equal, {Art. 146,) 
to twice the triangle, fbc; the triangle, a b d, and the parallelo- 
gram, b I, being both upon the same base, b d, and between the 
same parallels, 6 d and a I, the parallelogram, b I, is equal to twice 
the triangle, a b d ; the triangles,/ 6 c and a b d, being equal to 
one another, {Art. 141,) the square, g b, is equal to the parallelo- 
gram, b I, either being equal to twice the triangle, /6 c or a b d. 
The method of proving h c equal to c Z is exactly similar — thus 
proving the square, b e, equal to the squares, h c and g b, taken 
together. 

This problem, which is the 47th of the First Book of Euclid 
is said to have been demonstrated first by Pythagoras. It is sta 
led, (but the story is of doubtful authority,) that as a thank-oifer 
ing for its discovery he sacrificed a hundred oxen to the gods 
From this circumstance, it is sometimes called the hecatomb pro- 
blem. It is of great value in the exact sciences, more especially 
in Mensuration and Astionomy, in which many otherwise intri- 
cate calculations are by it made easy of solution. 

158. — Proposition. In a segment of a circle, the versed sine 
^equals the radius, less the square root of the difference of the 
squares of the radius and half-chord. That is, the versed sine, 
a c, {Fig. 109,) equals a h, less c h. ItTow ah is radius, hence 
tlie radius, minus c 5, equals a c, the versed sine. To find the 
value of c &, it will be observed that ch is the side of the square, 
cf, while the radius h d is the side of the square, h A, and the 
half-chord, c d, is the side of the square, c e ; also, that these 
three squares are made upon the three sides of the right angled 



AMERICAN HOUSE-CAKPENTEE. 



67 




Fig 109. 

triangle, h cd, and the square, h h, is therefore equal to the two 
squares, c e and g /*, {Art. 157 ;) therefore, the square, c f, is 
equal to the square, h A, minus the square, c e / — or, is equal to 
the difference of the squares onhd and c d. Consequently the 
square root of ef is equal to the square root of the difference 
of the squares on h d and g d; and since g h is the square root 
of of, therefore g h equals the square root of the difference of 
the squares onhd and g d — or, equals the square root of the 
difference of the squares of the radius and the half-chord. 
Haying found an expression for the value of c h, it remains 
merely to deduct this value from the radius, and the residue 
equals the versed sine ; for, as before stated, the versed sine, a g, 
equals the radius, a 5, minus oh ; therefore, the versed sine 
equals the radius, minus the square root of the difference of 
the squares on the radius and half-chord. The rule expressed 
algebraically is v=r--Vr'^—a'^ where v is the versed sine, r the 
radius, and a the half-chord. It is read, v equals r, minus the 
square root of the difference of the squares of r and a. 

159. — Proposition. In an equilateral octagon the semi- 
diagonal of a circumscribed square, having its sides coincident 
with four of the s'des of the octagon, equals the distance alon^ 



68 



PEACTICAL GEOMETEY 




Fig. 110. 



a Side of the square from its corner to the more remote angle 
of the octagon occurring on that side of the square. To prove 
this, it need only to be shown that the triangle, a o d^ {Fig. 
110,) is an isosceles triangle having its sides a o and a d^ equal. 
The octagon being equi-lateral, it is also equi-angular, therefore 
the angles, h c o^ e g o^ a d o^ &c., are all equal. Of the right- 
angled triangle, y^ c^fc oiAfe being equal, the two angles, fe c 
and fee are equal, {Art. 142,) and are therefore, {Art. 151,) 
each equal to half a right angle. In like manner it may be 
shown theit fa h and/ 5 a are also each equal to half a right 
angle. And since fee and f ah are equal angles, therefore 
the lines e c and a h are parallel, {Art. 135,) and hence the 
angles, e c o and a o d^ are equal. These being equal, and the 
angles e c o and ado being, by construction, equal, as before 
shown, therefore the angles a o d and ado are equal, and con- 
sequently the lines a o and a d are equal. {Art. 142.) 

160. — Projposition. An angle at the circumference of a 
circle is measured by half the arc that subtends it : that is, 
the angle ah c^ {Fig. Ill,) is equal to half the angle a d c. 
Through the centre, c?, draw the diameter, h e. The triangle 
ahdi^BXi isosceles triangle, a d and h d being radii, and there- 
fore equal ; hence the two angles, d ah and dh a^ are equal, 



A]VIERICAN HOUSE-CARPENTEJK. 




Fig. 111. 

{Art 142,) and the sum of these two angles is equal to the 
angle a d e^ {Art 149,) and therefore one of them, ah d^ is 
equal to the half oi a d e. The angles a d e and a h d {or 
a he) are both subtended by the arc a e. 'Now, since the angle, 
a d e,ie. measured by the arc a e, which subtends it, therefore 
the half of the angle, a d e, would be measured by the half 
of the arc a e; and since ah d is equal to the half of a d e, 
therefore ah d, or ah e, is measured by the half of the arc a e. 
It may be shown in like manner that the angle eh cis mea- 
sured by half the arc e c, and hence it follows that the angle, 
ah c, is measured by half the arc, a c, that subtends it. 

161. — Proposition. In a circle, aU the inscribed angles, 
a h G, {Fig. 112,) which stand upon the same side of the chord 
d e, are equal. For each angle is measured by half the arc 
df e^ {Art. 160,) hence the angles are aU equal. 

162. — Corollary. Equal chords, in the same circle, subtend 
equal angles. 

163. — Proposition. The angle formed by a chord and tan- 
gent is equal to any inscribed angle in the opposite segment 




70 



PRACnOAX GEOMETRY. 




Kff. 112. 




Fig. 113. 

of the circle ; that is, the angle Z>, {Fig. 113,) equals the angle 
A. Let cf be the chord, and a h the tangent ; draw the dia- 
meter, d c ; then ^ c 5 is a right angle, also <^ /* c is a right 
angle. {Art. 156.) The angles A and B together equal a 
right angle, {Art. 150 ;) also the angles B and B together 
equal a right angle, (equal the angle d ch ;) therefore the sum 
of A and B equals the sum of B and B. From each of these 
two equals, i;aking the like quantity B^ the remainders, A and 




AMERICAN HOUSE-CARPENTEK. 



71 



Z), are equal. Thus, it is proved for the angle at d / it is also 
true for any other angle ; for, since all other inscribed angles 
on that side of the chord line, <?/, equal the angle A^ {Art. 
161,) therefore the angle formed by a chord and tangent equals 
any angle in the opposite segment of the circle. This being 
proved for the acute angle, D, it is also true for the obtuse 




angle, a of; for, from any point, n, (Fig. 114:,) in the arc 
G nf^ draw lines to (^,y and g j now, if it can be proved t^^at 
the angle a g/ equals the angle f n c^ the entire proposition 
is proved, for the angle f n g equals any of all the inscribed 
angles that can be drawn on that side of the chord. {Art. 
161.) To prove, then, that a g/ equals g nf: the angle a cf 
equals the sum of the angles A and B / also the angle c nf 
equals the sum of the angles C and D. The angles B and i>, 
being inscribed angles on the same chord, df are equal. The 
angles G and A being right angles, {Art. 156,) are likewise 
equal. iN'ow, since A equals (7, and B equals D^ therefore 
the sum of A and B^ equals the sum of G and I) — or the angle 
a cf equals the angle c nf. 

164. — Proposition. Two chords, a h and g d^ {Fig- H^O 
intersecting, the parallelogram or rectangle formed by the two 
parts of one is equal to the rectangle formed by the two parts 
of the other. That is, g e m? Itiplied by e d, the product is 



72 



PBACTICAL GEOMETEY. 




Fig. 115. 



equal to the product oi a e multiplied \>j eh. The triangle A 
is similar to the triangle i?, because it has corresponding an- 
gles. The angle i equals the angle e, (Art. 148 ;) the angle at 
G equals the angle at a because they stand upon the same 
chord, d 5, {Art 161 ;) for the same reason the angle h equals 
the angle d^ for each stands upon the same chord, a c. There- 
fore, the triangle A having the same angles as the triangle B^ 
the length of the sides of one are in like proportion as the 
length of the sides in the other. So, e d : a e y, e h : c e. 
Hence, a e multiplied by ^ 5 is equal to e d mutiplied by c e — 
or the product of the means equals the product of the ex- 
tremes. 

165. — Proposition. In any circle, when a segment is given, 
the radius is equal to the sum of the squares of half the chord 
and of the versed sine, divided by twice the versed sine. Let 
a J, {Fig. 116,) be the chord line, and v the versed sine of the 
segment. By the preceding article the triangle A is shown to 
be like the triangle B^ having equal angles and proportionate 



AMEEICAN HOUSE-CAEPENTEE. 



T3 




length of sides. Therefore, v : n::m : i, or — =zi; that is, i is 

equal to the square of n {orn x n) divided by v. This result 
being added to v equals the diameter o a?, which may be indi 

cated by the letter d : thus, hv = i-hv = d: and the half 

of this, or — - — =z — =r = the radius. Reducing this expres- 
sion by multiplying the numerator and denominator each by the 
like quantity, viz. -y, there results, — = r ; and where c 

represents the chord, the expression is, — = r : that is, 

as stated above, Ihe radius is equal to the sum of the squares 
of half the chord and of the versed sine, divided by twice the 
versed sine. 

166. — ProposiUon, Any ordinate, m n, (-?%. HT,) in the 
segment of a circle, is equal to the square root of the difference 

10 



ri 



PEACTICAL GEOMETEY. 




of the squares of the radius and abscissa, {d n,) less the differ- 
ence of the radius and versed sine. So, if the chord a h, and 
the versed sine c d, be given, the length of any number of 
ordinates may be found by which to describe the arc. Find 
the radius, c e, by the preceding Article. It will be observed 
that e m is also radius. Then, to find the length of the ordi- 
nate, m n, make e o equal to d n : now, according to Article 
157, the square of e o taken from the square of e m, the residue 
equals the square of o m, and the square root of this residue 
will be the length of the line o m. Then from o m take o n 
equal to e d^ and the result will be the length of m n. That is, 
the ordinate is equal to the square root of the difference of the 
squares of the radius and abscissa, less the difference of the 
radius and versed sine. This may be expressed algebraically 
thus : y = -x/r"^ — x" — {r — v), where y is the ordinate, r the 
radius, x the abscissa, and v the versed sine ; — d n being the 
abscissa of the ordinate n m, d g the abscissa of the ordinate 



AMERICAN HOUSE-CAEPENTEE. 75 

gf^ &c. : tlie abscissa being in each case the distance from the 
foot of the versed sine, c d^ to the foot of the ordinate whose 
length is sought. 




167. — Proposition. The sides of any quadrangle being 
bisected, and lines drawn joining the points of bisection in the 
adjacent sides, these lines will form a parallelogram. Draw 
the diagonals, a l and c d^ {Fig. 118.) It will here be per- 
ceived that the two triangles, a e o and a g d^ are homologous, 
having like angles and proportionate sides. Two of the sides 
of one triangle lie coincident with the two corresponding sides 
of the other triangle, therefore the contained angles between 
these sides in each triangle are identical. By construction, 
these corresponding sides are proportionate ; a c being equal 
to twice a e^ and a d being equal to twice a o ; therefore the 
remaining sides are proportionate, c d being equal to twice e <?, 
hence the remaining corresponding angles are equal. Since, 
then, the angles a e o and a g d are equal, therefore the line e o 
is parallel with the diagonal c d — so, likewise, the line mnh 
parallel to the same diagonal, g d. If, therefore, these two 
lines, e o and m n, are parallel to the same line, g d, they must 
be parallel to each other. In the same manner the lines o n 
and e m are proved parallel to the diagonal, a 5, and to each 



76 



PEACTICAL GEOMETRY. 



Other ; therefore the inscribed figure, m e o n, is sl parallelo- 
gram. It may be remarked also, that the parallelogram so 
formed will contaii just one-half the area of the circumscribing 
quadrangle. 



These demonstrations, which relate mostly to the problems 
previously given, are introduced to satisfy the learner in regard 
to their mathematical accuracy. By studying and thoroughly 
understanding them, he will soonest arrive at a knowledge of 
their importance, and be likely the longer to retain them in 
memory. Should he have a reUsh for such exercises, and wish 
to continue them farther, he may consult Euclid's Elements, in 
which the whole subject of theoretical geometry is treated of 
in a manner sufficiently intelligible to be understood by the 
young mechanic. The house-carpenter, especially, needs infor- 
mation of this kind, and were he thoroughly acquainted with 
the principles of geometry, he would be much less liable to 
commit mistakes, and be better qualified to excel in the execu- 
tion of his often difficult undertakings. 



SECTION Jl.— ARCHITECTTTRE. 



HISTORY OF ARCHITECTURE. 

168. —Architecture has been defined to be — " the art of build- 
ing ;" but, in its common acceptation, it is — " the art of designing 
and constructing buildings, in accordance with such principles as 
constitute stability, utility and beauty." The hteral signification 
of the Greek word archi-tecton, from which the word architect 
is derived, is chief-carpenter ; but the architect has always been 
known as the chief designer rather than the chief builder. Of 
the three classes into which architecture has been divided — viz., 
Civil, Military, and Naval, the first is that which refers to ihe 
construction of edifices known as dwellings, churches and other 
public buildings, bridges, &c., for the accommodation of civilized 
man — and is the subject of the remarks which follow. 

169. — This is one of the most ancient of the arts : the scrip- 
tures inform us of its existence at a very early period. Cain, 
the son of Adam, — " builded a city, and called the name of the 
city after the name of his son, Enoch'' — but of the peculiar style 
or manner of building we are not informed. It is presumed that 
it was not remarkable for beauty, but that utility and perhaps sta- 
bility were its characteristics. Soon after the deluge — that me 



78 AMERICAN HOUSE-CARPENTER. 

morable event, which removed from existence all traces of the 
works of man — the Tower of Babel was commenced. This was 
a work of such magnitude that the gathering of the materials, 
according to some writers, occupied three years ; the period from 
its commencement until the work was abandoned, was twenty- 
two years ; and the bricks were like blocks of stone, being twenty 
feet long, fifteen broad and seven thick. Learned men have given 
it as their opinion, that the tower in the temple of Belus at Baby- 
lon was the same as that which in the scriptures is called the 
Tower of Babel. The tower of the temple of Belus was square 
at its base, eacn side measuring one lurlong, and consequently 
half a mile in circumference. Its form was that of a pyramid 
and its height was 660 feet. It had a winding passage on the 
outside from the base to the summit, which was wide enough for 
two carriages. 

170. — Historical accounts of ancient cities, of which there are 
now but few remains — such as Babylon. Palmyra and Ninevah 
of the Assyrians ; Sidon, Tyre, Aradus and Serepta of the Phoe- 
nicians ; and Jerusalem, with its splendid temple, of the Israelites 
— show that architecture among them had made great advances. 
Ancient monuments of the art are found also among other nations ! 
the subterraneous temples of the Hindoos upon the islands, Ele- 
phanta and Salsetta ; the ruins of Persepolis in Persia ; pyramids, 
obelisks, temples, palaces and sepulchres in Egypt — all prove that 
the architects of those early times were possessed of skill and 
judgment highly cultivated. The principal characteristics of 
their works, are gigantic dimensions, immoveable solidity, and, in 
some instances, harmonious splendour. The extraordinary size 
of some is illustrated in the pyramids of Egypt. The largest of 
these stands not far from the city of Cairo : its base, which is 
square, covers about 11| acres, and its height is nearly 500 feet 
The stones of which it is built are immense — the smallest being 
full thirty feet long. 

171 .--Among the Greeks, architecture was cultivated as a fine 



ARCHITECTURE. 79 

art, and rapidly advanced towards perfection. Dignity and grace 
were added to stability and magnificence. In the Doric order, 
their first style of building, this is fully exemplified. Phidias, 
Ictinus and Gallicrates, are spoken of as masters in the art at this 
period: the encouragement and support of Pericles stimulated 
them to a noble emulation. The beautiful temple of Minerva, 
erected upon the acropolis of Athens, the Propyleum, the Odeum 
and others, were lasting monuments of their success. The Ionic 
and Corinthian orders were added to the Doric, and many mag- 
nificent edifices arose. These exemplified, in their chaste propor- 
tions, the elegant refinement of Grecian taste. Improvement in 
Grecian architecture continued to advance, until perfection seems 
to have been attained. The specimens which have been partially 
preserved, exhibit a combination of elegant proportion, dignified 
simplicity and majestic grandeur. Architecture among the 
Greeks was at the height of its glory at the period immediately 
preceding the Peloponnesian war; after which the art declined. 
An excess of enrichment succeeded its former simple grandeur ; 
yet a strict regularity was maintained amid the profusion of orna- 
ment. After the death of Alexander, 323 B. C, a love of gaudy 
splendour increased : the consequent decline of the art was 
visible, and the Greeks afterwards paid but little attention to the 
science. 

172. — While the Greeks were masters in architecture, whicH 
they applied mostly to their temples and other public buildings, 
the Romans gave their attention to the science in the construction 
of the many aqueducts and sewers with which Rome abounded ; 
building no such splendid edifices as adorned Athens. Corinth 
and Ephesus. until about 200 years B. C, when their intercourse 
with the Greeks be-came more extended. Grecian architecture 
was introduced into Rome by Sylla ; by whom, as also by Marius 
and Caesar, many large edifices were erected in various cities of 
Italy. But under Caesar Augustus, at about the beginning of the 
christian era, tlie art arose to the greatest perfection it ever at- 



80 AMERICAN HOUSE-CARPENTER. 

tained in Italy. Under his patronage. Grecian artists were en- 
couraged, and many emigrated to Rome. It was at about this 
time that Solomon's temple at Jerusalem was rebuilt by Herod — 
a Roman. This was 46 years in the erection, and was most pro- 
bably of the Grecian style of building — perhaps of the Corin- 
thian order. Some of the stones of which it was built were 46 
feet long, 21 feet high and 14 thick ; and others were of the 
astonishing length of 82 feet. The porch rose to a great height ; 
the whole being built of white marble exquisitely polished. This 
IS the building concerning which it was remarked — "Master, see 
what manner of stones, and what buildings are here." For the 
construction of private habitations also, finished artists were em- 
ployed by the Romans : their dwellings being often built with the 
finest marble, and their villas splendidly adorned. After Augus- 
tus, his successors continued to beautify the city, until the reign of 
Constantine; who, having removed the imperial residence to 
Constantinople, neglected to add to the splendour of Rome ; and 
the art, in consequence, soon fell from its high excellence. 

Thus we find that Rome was indebted to Greece for what she 
possessed of architecture — not only for the knowledge of its prin- 
ciples, but also for many of the best buildings themselves ; these 
having been originally erected in Greece, and stolen by the un- 
principled conquerors — taken down and removed to Rome. 
Greece was thus robbed of her best monuments of architecture. 
Touched by the Romans, Grecian architecture lost much of its 
elegance and dignity. The Romans, though justly celebrated 
for their scientific knowledge as displayed in the construction of 
their various edifices, were not capable of appreciating the simple 
grandeur, the refined elegance of the Grecian style ; but sought 
to improve upon it by the addition of luxurious enrichment, and 
thus deprived it of true elegance. In the days of Nero, whose 
palace of gold is so celebrated, buildings were lavishly adorned. 
Adrian did much to encourage the art ; but not satisfied with the 
simplicity of the Grecian style, the artists of his time aimed at 



ARCHITECTURE. 81 

inventing new" ones, and added to the already redundant embel- 
lishments of the previous age. Hence the origin of the pedestal, 
the great variety of intricate ornaments, the convex frieze, the 
round and the open pediments, &c. The rage for luxury 
continued until Alexander Severus, who made some impDre- 
ment ; but very soon after his reign, the art began rapidly to 
decline, as particularly evidenced in the mean and trifling charac- 
ter of the ornaments. 

173. — The Goths and Vandals, when they overran the coun- 
tries of Italy, Greece, Asia and Africa, destroyed most of the 
works of ancient architecture. Cultivating no art but that of 
war, these savage hordes could not be expected to take any interest 
in the beautiful forms and proportions of their habitations. P^rom 
this time, architecture assumed an entirely different aspect. The 
celebrated styles of Greece were unappreciated and forgotten ; and 
modern architecture took its first step on the platform of existence. 
The Goths, in their conquering invasions, gradually extended it 
over Italy, France, Spain, Portugal and Germany, into England. 
From the reign of Gallienus may be reckoned the total extinction 
of the arts among the Romans. From his time until the bth or 
7th century, architecture was almost entirely neglected. The 
buildings which were erected during this suspension of the arts, 
were very rude. Being constructed of the fragments of the edi- 
fices which had been demolished by the Visigoths in their unre- 
strained fury, and the builders being destitute of a proper know- 
ledge of architecture, many sad blunders and extensive patch- 
work might have been seen in their construction — entablatures 
inverted, columns standing on their wrong ends, and other ridi- 
culous arrangements characterized their clumsy work. The vast 
number of columns which the ruins around them afforded, they 
used as piers in the construction of arcades — which by some is 
thought, aft(ir having passed through various changes, to have 
been the origin of the plan of the Gothic cathedral. Buildings 
generally, which ar3 not of the classical styles, and which were 

11 



8'J AMERICAN HOUSE-CARPENTER. 

erected after the fall of the Roman empire, have by some been 
indiscrimmately mcluded under the term Gothic. But the 
changes which architecture underwent during the dark ages, show 
that there were several distinct modes of building. 

iY4. — Theodoric, king of the Ostrogoths, a friend of the arts, 
who reigned in Italy from A. D. 493 to 525, endeavoured to re- 
store and preserve some of the ancient buildings ; and erected 
others, the ruins of which are still seen at Yerona and Ravenna. 
Simplicity and strength are the characteristics of the structures 
erected by him ; they are, however, devoid of grandeur and ele- 
gance, or fine proportions. These are properly of the Gothic 
style ; by some called the old Gothic to distinguish it from the 
pointed style, which is generally called moderji Gothic. 

175. — The Lombards, who ruled in Italy from A. D. 568, had 
no taste for architecture nor respect for antiquities. Accordingly, 
they pulled down the splendid monuments of classic architecture 
which they found standing, and erected in their stead huge build- 
ings of stone Avhich were greatly destitute of proportion, elegance 
or utility — their characteristics being scarcely any thing more than 
stability and immensity combined with ornaments of a puerile cha- 
racter. Their churches were disfigured with rows of small columns 
along the cornice of the pediment, small doors and windows with 
circular heads, roofs supported by arches having arched buttresses 
to resist their thrust, and a lavish display of incongruous orna- 
ments. This kind of architecture is called, the Lombard style, 
and was employed in the 7th century in Pavia, the chief city of 
the Lombards; at which city, as also at many other places, a 
great many edifices were erected in accordance with its inelegant 
forms. 

176. — The Byzantine architects, from Byzantium, Constantino- 
ple, erected many spacious edifices ; among which are included 
the cathedrals of Bamberg, Worms and Mentz, and the most an 
cient part of the minster at Strasburg ; in all of these they com- 
bined the Roman-Ionic order with the Gothic of the Lombards. 



ARCHITECTURE. 83 

This stj'le is called the Lombard-Byzantine. To the last style 
there were afterwards added cupolas similar to thost used in the 
east, together with numerous slender pillars with tasteless capi- 
tals, and the many minarets which are the characteristics of the 
proper Byzantine^ or Oriental style. 

17 7. — In the eighth century, when the Arabs and Moors de- 
stroyed the kingdom of the Goths, the arts and sciences were 
mostly in possession of the Musselmen-conquerors ; at which 
time there were three kinds of architecture practised ; viz : the 
Arabian, the Moorish and the modern-Gothic. The Arabian 
style was formed from Greek models, having circular arches 
added, and towers which terminated with globes and minarets. 
The Moorish is very similar to the Arabian, being distinguished 
from it by arches in the form of a horse-shoe. It originated in 
Spain in the erection of buildings with the ruins of Roman archi- 
tecture, and is seen in all its splendour in the ancient palace of the 
Mohammedan monarchs at Grenada, called the Alhambra, or red- 
house. The Modern-Gothic was originated by the Visigoths 
in Spain by a combination of the Arabian and Moorish styles ; 
and introduced by Charlemagne into Germany. On account of 
the changes and improvements it there underwent, it was, at about 
the 13th or 14th century, termed the German^ or romantic style. 
It is exhibited in great perfection in the towers of the minster of 
Strasburgh, the cathedral of Cologne and other edifices. The 
most remarkable features of this lofty and aspiring style, are the 
lancet or pointed arch, clustered pillars, lofty towers and flying 
buttresses. It was principally employed in ecclesiastical archi- 
tecture, and in this capacity introduced into France, Italy, Spain, 
and England. 

178. — The Gothic architecture of England is divided into the 
Norman^ the Early-English^ the Decorated^ and the Perpen- 
dicular styles. The Norman is principally distinguished by the 
character of its ornaments — the chevron, or zigzags being the 
most common. Buildings in this style were erected in the 12th 



84- AMERICAN HOUSE-CARPEXTER. 

century. The Early-English is celebrated for the beauty of its 
edifices, the chaste simplicity and purity of design which they 
display, and the pecuharly graceful character of its foliage. This 
style is of the 18th century. The Decorated style, as its name 
implies, is charLit^rized by a great profusion of enrichment, 
Avhicli consists principally of the crocket, or feathered-ornament, 
and ball-flower. It was mostly in use m the 14th century. The 
Perpendicular style, which dates from the 15th century, is distin- 
guished by its high towers, and parapets surmounted with spires 
similar in number and grouping to oriental minarets. 

179. — Thus these several styles, which have been erroneously 
termed Gothic^ were distinguished by peculiar characteristics as well 
as by different names. The first symptoms of a desire to return to a 
pure style in architecture, after the ruin caused by the Goths, was 
manifested in the character of the art as displayed in the church 
of St. Sophia at Constantinople, which was erected by Justinian 
in the 6th century. The church of St. Mark at Venice, which 
arose in the 10th or 11th century, was the work of Grecian archi- 
tects, and resembles in magnificence the forms of ancient archi- 
tecture. The cathedral at Pisa, a wonderful structure for the age, 
was erected by a Grecian architect in 1016. The marble with 
which the walls of this building were faced, and of which the four 
rows of columns that support the roof are composed, is said to be 
of an excellent character. The Campanile, or leaning-tower as it 
is usually called, was erected near the cathedral in the 15th cen- 
tury. Its inclination is generally supposed to have arisen from 
a poor foundation ; although by some it is said to have been thus 
constructed originally, in order to inspire in the minds of the 
beholder sensations of sublimity and awe. In the 13th century, 
the science in Italy was slowly progressing ; many fine churches 
v/ere erected, the style of which displayed a decided advance in 
the progress towards pure classical architecture. In other parts 
of Europe, the Gothic, or pointed style, was prevalent. The 
cat], -^-.Iral at Strajsburg, designed ' y Irwin Steinbeck, was erected 



AJECHTTECTUEE. 85 

in the 13th and 14th centuries. In France and England dur- 
ing the 14th century, many very superior edifices were erected 
in this style. 

180. — In the 14th and 15th centuries, and particularly in the 
latter, architecture in Italy was greatly revived. The masters 
began to study the remains of ancient Eoman edifices ; and many 
splendid buildings were erected, which displayed a purer taste 
in the science. Among others, St. Peter's of Rome, which was 
built about this time, is a lasting monument of the architecturpJ 
skill of the age. Giocondo, Michael Angelo, Palladio, Yignola, 
and other celebrated architects, each in their tm^n, did much to 
restore the art to its former excellence. In the edifices which 
were erected under their direction, however, it is plainly to be 
seen that they studied not from the pure models of Greece, but 
from the remains of the deteriorated architecture of Eome. The 
high pedestal, the coupled columns, the rounded pediment, the 
many curved-and-twisted enrichments, and the convex frieze, 
were unknown to pure Grecian architecture. Yet their efforts 
were serviceable in correcting, to a good degree, the very 
impure taste that had prevailed since the overthrow of the Eo- 
man empire. 

181. — At about this time, the Italian masters and numerous 
artists who had visited Italy for the pui*pose, spread the Eoman 
style over various countries of Europe ; which was gradually re- 
ceived into favor in place of the modern-Gothic. This fell into 
disuse ; although it has of late years been again cultivated. It 
requires a building of great magnitude and complexity for a per- 
fect display of its beauties. In America, the pure Grecian style 
was at first more or less studied ; and perhaps the simplicity of 
its principles would be better adapted to a republican country, 
than the intricacy and extent of those of the Gothic ; but at the 
present time the latter style is being introduced, especially for 
eccJesiastical structures. 



86. AMERICAN HOUSE-CARPENTER. 

STYLES OF ARCHITECTURE. 

1 82. — It is generally acknowledged that the various styles in 
architecture, were originated in accordance with the different 
pursuits of the early inhabitants of the earth ; and were brought 
by their descendants to their present state of perfection, througlr 
the propensity for imitation and desire of emulation which are 
found more or less among ail nations. Those that followed 
agricultural pursuits, from being employed constantly upon 
the same piece of land, needed a permanent residence, and the 
wooden hut was the offspring of their wants ; while the shep- 
herd, who followed his flocks and was compelled to traverse 
large tracts of country for pasture, found the tent to be the 
most portable habitation ; again, the man devoted to hunting 
and fishing — an idle and vagabond w^ay of living — ^is naturally 
supposed to have been content with the cavern as a place of 
shelter. The latter is said to have been the origin of the 
Egyptian style; while the curved roof of Chinese structures 
gives a strong indication of their having had the tent for their 
model ; and the simplicity of the original style of the Greeks, 
(the Doric,) shows quite conclusively, as is generally conceded, 
that its original was of wood. The modern-Gothic, or pointed 
style, which was most generally confined to ecclesiastical 
structures, is said by some to have originated in an attempt to 
imitate the bower, or grove of trees, in which the ancients per- 
formed their idol-worship. 

183. — There are numerous styles, or orders, in architecture; 
and a knowledge of the peculiarities of each is important to the 
student in the art. An Order, in architecture, is composed of 
three principal parts, viz : the Stylobate, the Column and the 
Entablature. 

184. — ^The Stylobate is the substructure, or basement, upon 
which the columns of an order are arranged. In Roman archi- 
tecture — especially in the interior of an edifice — it frequently 
occurs that each column has a separate substructure ; this is 



AKCHITECTUEE. 87 

called 2i pedestal. If possible, the pedestal should be avoided 
in all cases; because it gives to the column, the appearance of 
having been originally designed for a small building, and after- 
wards pieced-out to make it long enough for a larger one. 

185. — The Colu:mn is composed of the base, shaft and capital. 

186. — ^The EntablatuFvE, above and supported by the co- , 
lumns, is horizontal ; and is composed of the architrave, frieze V 
and cornice. These principal parts are again divided into '] 
vavious members and mouldings. (See Sect. III.) 

187. — ^The Base of a column is so called from hasis^ a found- 
ation, or footing. 

188. — The Shaft, the upright part of a column standing 
upon the base and crowned with the capital, is from shafto, to 
dig — in the manner of a well, whose inside is not unlike the 
form of a column. 

189. — ^The Capital, from hejpliale or ca])ut^ the* head, is the 
UDpermost and crowning part of the column. 

190. — The Architkave, from archly chief or principal, and 
trahs^ a beam, is that part of the entablature which lies in 
immediate connection with the column. * 

191. — The Fkieze, from Jibron, sl fringe or border, is that 
part of the entablature which is immediately above the archi- 
trave and beneath the cornice. It was called by some of the ^ 
ancients, zophorus, because it was usually enriched with sculp- 
tured animals. 

192. — ^The Cornice, from corona, sl crown, is the upper and 
projecting part of the entablature — being also the uppermost 
and crowning part of the whole order. 

193. — ^The Pediment, above the entablature, is the triangular 
portion which is formed by the inclined edges of the roof at ' 
the end of the building. In Gothic architecture, the pediment . 
is called, a gable. 

191. — The Tympanum is the perpendicular triangular surface 
which is enclosed by the c<! mice of the pediment. 



88 AMERICAN HOrSE-CABPENTEE. 

195. — The Attic is a small order, consisting of pilasters and 
entablature, raised above a I'Arger order, instead of a pediment. 
An attic story is tbe upper stoiy, its windows being usually 
square. 

196. — An order, in architecture, has its several parts and 
members proportioned to one another by a scale of 60 equal 
parts, which are called minutes. If the height of buildings 
were always the same, the scale of equal parts would be a 
fixed quantity — an exact number of feet and inches. But as 
buildings are erected of different heights, the column and its 
accompaniments are required to be of different dimensions. 
To ascertain the scale of equal parts, it is necessary to know 
the height to which the whole order is to be erected. This 
must be divided by the number of diameters which is directed 
for the order under consideration. Then the quotient obtained 
by such division, is the length of the scale of equal parts — and 
is, also, the diameter of the column next above the base. For 
instance, in the Grecian Doric order the whole height, includ- 
ing coluiim and entablature, is 8 diameters. Suppose now it 
were desirable to construct an example of this order, forty feet 
liigh. Then 40 feet divided by 8, gives 5 feet for the length 
of the scale ; and this being divided by 60, the scale is com- 
pleted. The upright columns of figures, marked H and P^ by 
the side of the drawings illustrating the orders, designate the 
height and the projection of the members. The projection of 
veach member is reckoned from a line passing through the axis 
of the column, and extending above it to the top of the enta- 
blature. The figures represent minutes, or 60ths, of the major 
diameter of the shaft of the column. 

197. — Grecian Styles. The original method of building 
among the Greeks, was in what is called the Doric order : to 
this were afterwards added the Ionic and the Corinthian, 
These three were the only styles known among them. Each is 
distingrishr-i from the other two, by not only a peculiarity of 



AliCHITECTlIRE. 



89 



some one or more of its principal parts, but also "by a particular 
destination. The character of the Doric is robust, manly and 
Herculean-like ; that of the Ionic is more delicate, feminine, 
matronly ; while that of the Corinthian is extremely delicate, 
youthful and virgin-like. However they may differ in their 
general character, they are alike famous for grace and dignity, 
elegance and grandeur, to a high degree of perfection. 

198. — The Dome Ordee, {Fig. 120,) is so ancient that its origin 
is unknown — although some have pretended to have discovered 
it. But the most general opinion is, that it is an improvement 
upon the original wooden buildings of the Grecians. These no 
doubt were very rude, and perhaps not unlike the following 
fiffure. 







Fig. 119. 

The trunks of trees, set perpendicularly to support the roof, 
may be taken for columns ; the tree laid upon the tops of the 
perpendicular ones, the architrave ; the ends of the cross-beams 
which rest upon the architrave, the triglyphs ; the tree laid on 
the cross-beams as a support for the ends of the rafters, the 
bed-moulding of the cornice ; the ends of the rafters which 
project beyond the bed-moulding, the mutules; and perhaps 
the projection of the roof in front, to screen the entrance from 
the weather, gave origin to the portico. 

The peculiarities of the Doric order are the triglyphs — those 

parts of the frieze which have perpendicular channels cut in 

12 



DORIC ORDER. 




Fig. 120 



AKCmTECTUEE. 91 

their surface ; the absence of a base to the cokimn — as alco of 
fillets between the flutings of the column, and the plainness of 
the capital. The trigljphs are to be so disposed that the width 
of the metopes — the spaces between the triglyphs — shall be 
equal to their height. 

199. — The inUrcolximniation^ or space between the columns, 
is regulated by placing the centres of the columns under the 
centres of the trigljphs — except at the angle of the building ; 
where, as may be seen in Fig. 120, one edge of the triglyph 
must be over the centre of the column.* Where the columns 
are so disposed that one of them stands beneath every other tri- 
glyph, the arrangement is called, 7nono-triglyph, and is most 
common. When a column is placed beneath every third tri- 
glyph, the arrangement is called diastyle ; and when beneath 
every fourth, aroeostyle. This last style is the worst, and is sel- 
dom adopted. 

200. — ^The I)oric order is suitable for buildings that are des- 
tined for national purposes, for banking-houses, &c. Its ap- 
pearance, though massive and grand, is nevertheless rich and 
graceful. The Patent Office at Washington, and the Custom- 
House at New York, are good specimens of this order. 



* GaEciAN Doric Order. When the width to be occupied by the whole front is limited; to deter- 
mine the diameter of the column. 
The relation between the parts may be expressed thus : 

60^ 

^~rf(6 + c) + (60 — c) 
Where a equals the width in feet occupied by the columns, and their intercolumniations takec 
collectively, measured at the base ; b equals the width of the metope, in minutes ; c equals the width 
of the triglyphs in minutes ; d equals the number of metopes, and x equals the diameter in feet. 

Example.— A front of six columns— hexastyle— 61 feei wide ; the frieze having one triglyph over 
each intercoliimniation, or mono-triglyph. In this case, tliere being five intercolumniations and two 
metopes over each, therefore there are 5 X 2= 10 metopes. Let the metope equal 42 minutes and 
the triglyph equal ^S. Then a = 6 1 ; Z> = 42 ; c = 28 ; and <i = 10 ; and the formula above becomes, 

— 60 X 61 60 X 61 3660 

"■ - 10(4-i + 2Sr+"(00-28) "= mao + ¥2= 132= ^ ^""^^ = ^^'^ diameter n" juired. 

Exa7nple.—An octastyle front, 8 columns, 184 feet wide, three metopes over each intercolurani* 
tion, 21 in all, and the metope and triglyph 42 am. 28, as before. Then, 

60X184 11040 „n 

"^ - 2U42 + 28H-l603:^ = 1502 = '■^•'t Jo 2 ^«^' ^ ^^« diameter required. 



92 



UmiQ OEDlilR. 




Fig. 121. 



AKCHITECTPEE. 93 

201.— Tlie loxic Oeder. (Fig. 121.) Tlie Doric was for 
some time the only order in use among the Greeks. They gave 
their attention to the cultivation of it, until perfection seems to 
have been attained. Their temples were the principal objects 
upon which their skill in the art was displayed ; and as the 
Doric order seems to have been well fitted, by its massive pro- 
portions, to represent the character of their male deities rather 
than the female, there seems to have been a necessity for an- 
other style which should be emblematical of feminine graces, 
and with which they might decorate such temples as were de- 
dicated to the goddesses. Hence the origin of the Ionic order. 
This was invented, according to historians, by Hermogenes of 
Alabanda ; and he being a native of Caria, then in the posses- 
sion of the lonians, the order was called, the Ionic. 

202. — The distinguishing features of this order are the vo 
lutes^ or spirals of the capital ; and the dentils among the bed- 
mouldings of the cornice : although in some instances, dentils 
are wanting. The volutes are said to have been designed as a 
representation of curls of hair on the head of a matron, of whom 
the whole column is taken as a semblance. 

203. — ^The intercolumniation of this and the other orders — 
both Roman and Grecian, with the exception of the Doric- 
are distinguished as follows. "When the interval is one and a 
half diameters, it is called, pycnostyle^ or columns thick-set ; 
when two diameters, systyle / vrhen two and a quarter diame- 
ters, eustyle / when three diameters, diastyle / and when more 
than three diameters, aroeostyle^ or columns thin-set. In all the 
orders, when there are four columns in one row, the arrange- 
ment is called, tetrastyle ; when there are six in a row, hexa- 
style / and when eight, octastyle. 

204. — The Ionic order is appropriate for churches, colleges, 
seminaries, libraries, all edifices dedicated to literature and the 
arts, and all places of peace and tranquillity. The front of the 



94 



ARCHITECnniE- 



Mercliauts' Exchange, ]^ew York city, is a good fcjpecimen of 
tliis order. 

205. — To describe the Ionic volute. Draw a perpendicular 
from a to 5, {Fig. 122,) and make a s equal to 20 min. or to 4 
of ttie whole height, a c ; draw 5 0, at right angles to s a^ and 
equal to 1 J min. ; upon 0^ with 2 J min. for radius, describe the 
eje of the volute ; about (?, the centre of the eje, di-aw the 
square, r ^ 1 2, with sides equal to half the diameter of the 
eye, viz. 2^ min., and divide it into 144 equal parts, as shown 




AMEEICAlf HOUSE-CAEPENTER. 



95 




Fig. 123. 



at Fig. 123. The several centres in rotation are at the angles 
formed bj the heavy lines, as figured, 1, 2, 3, 4, 5, 6, &c. The 
position of these angles is determined by commencing at the 
point, 1, and making each heavy line one part less in length 
than the preceding one. ]^o. 1 is the centre for the arc, a 5, 
{Fig. 122;) 2 is the centre for the arc, l c ; and so on to the 
last. The inside spiral line is to be described from the centres, 
a?, a?, cc, &c., {Fig. 123,) being the centre of the first small 
square towards the middle of the eye from the centre for the 
outside arc. Tlie breadth of the fillet at a j^ is to be made 
equal to 2tV min. This is for a spiral of three revolutions ; but 
one of any number of revolutions, as 4 or 6, may be drawn, by 
dividing of, {Fig. 123,) into a corresponding number of equal 
parts. Then divide the part nearest the centre, <?, into two 
parts, as at h ; join o and 1, also o and 2; draw h 3, parallel 
to {> 1, and h 4, parallel to <? 2 ; then the lines, c> 1, (9 2, A 3, A 4, 
will determine the length of the heavy hues, and the place of 
the centres. (See Art. 489.) 



9G 



AMERICAN HOUSE-CAKPENTER. 



206. — ^The Corinthian Order, {Fig. 125,) is in general like 
the Ionic, tliongh the proportions are lighter. The Corinthian 
displays a more airy elegance, a richer appearance ; but its 
distinguishing feature is its beautiful capital. This is gene- 
rally supposed to have had its origin in the capitals of the 
columns of Egyptian temples ; which, though not approaching 
it in elegance, have yet a similarity of form with the Corin- 
thian. The oft-repeated story of its origin which is told b}^ 
Yitruvius — an architect who flourished in Rome, in the days 
of Augustus Csesar — though pretty generally considered to be 
fabulous, is nevertheless worthy of being again recited. It is 
this : a young lady of Corinth was sick, and finally died. 
Her nurse gathered into a deep basket, such trinkets and 
keepsakes as the lady had been fond of when alive, and 
placed them upon her grave ; covering the basket with a flat 
stone or tile, that its contents might not be disturbed. The 
basket was placed accidentally upon the stem of an acanthus 
plant, which, shooting forth, enclosed the basket with its foli- 
age ; some of which, reaching the tile, turned gracefully over 
in the form of a volute. 

A celebrated sculptor, Calima- 
chus, saw the basket thus deco- 
rated, and from the hint which it 
suggested, conceived and con- 
structed a capital for a column. 
This was called Corinthian from 
the fact that it was invented and 
first made use of at Corinth. 
207. — The Corinthian being the gayest, the richest, and 
most lovely of all the orders, it is appropriate for edifices 
which are dedicated to amusement, banqueting and festiv- 
ity — for all places where delicacy, gayety and splendour are 
desirable. 

208. — In addition to the three regular orders of architecture, 




I-'. jr. vn. 



ARCHITECTURE. 



f^r?^ 




CORINTHIAN ORDER.-Fig. 125. 

13 



9S AMEBIC a:^^ house-carpentek. 

it was sometimes customary among the Greeks — and after , 
wards among otlier nations — to employ representations of the 
human form, instead of columns, to support entablatures ; these 
were called Persians and Caryatides. 

209. — Persians are statues of men, and are so called in 
commemoration of a victory gained over the Persians by Pau- 
sanias. The Persian prisoners were brought to Athens and 
condemned to abject slavery; and in order to represent them 
in the lowest state of servitude and degradation, the statues 
were loaded with the heaviest entablature, the Doric. 

210. — Caryatides are statues of women dressed in long 
robes after the Asiatic manner. Their origin is as follows. 
In a war between the Greeks and the Gary ans,. the latter were 
totally vanquished, their male population extinguished, and 
their females carried to Athens. To perpetuate the memory 
of this event, statues of females, having the form and dress of 
the Caryans, were erected, and crowned with the Ionic or Co- 
rinthian entablature. The caryatides were generally formed 
of about the human size, but the persians much larger ; in 
order to produce the greater awe and astonishment in the 
beholder. The entablatures were proportioned to a statue in 
like manner as to a column of the same height. 

211. — ^These semblances of slavery have been in frequent 
use among moderns as well as ancients ; and as a relief from 
the stateliness and formality of the regular orders, are capable 
of forming a thousand varieties ; yet in a land of liberty such 
marks of human degradation ought not to be perpetuated. 

212. — PoMAN Styles. Strictly speaking, Pome had no 
:architecture of her own — all she possessed was borrowed from 
other nations. Before the Pomans exchanged intercourse 
with the Greeks, they possessed some edifices of considerable 
extent and merit, which were erected by architects from Etru- 
ria; but Pome was principally indebted to Greece for what 
she acquired of the art. Although there is no such thing as 



ARCHITECTUEE. 



99 






? |j.l^..^Jfe^J^J.ifeH^^A:^^^ 




^ 



i5§K^l ryl r^ P 



26 



LIULAUUIA 






29^2-. 



fVl l\71 P 



5'ir^-^ 



4-4 



"f fi" ,r ^" r * f/' i'V 1'^^ t' ,'i I'- w I" "• IK n '■' iT 





Fig. 126. 



100 AI^IEPwICAN HOUSE-CAEPEXTEK. 

an architecture of Eoman invention, yet no nation, perhaps, 
ever was so devoted to the cultivation of tlie art as the Eo- 
man. Whether we consider the number and extent of their 
structures, or the lavish richness and splendour with which 
they were adorned, we are compelled to yield to them our 
admiration and praise. At one time, under the consuls and 
emperors, Rome employed 400 architects. The public works 
— such as theatres, circuses, baths, aqueducts, &c. — were, in 
extent and grandeur, beyond any thing attempted in modern 
times. Aqueducts were built to convey water from a distance 
of 60 miles or more. In the prosecution of this work, rocks 
and mountains were tunnelled, and valleys bridged. Some of 
the latter descended 200 feet below the level of the water; 
and in passing them the canals were supported by an arcade, 
or succession of arches. Public baths are spoken of as large 
as cities ; being fitted up with numerous conveniences for 
exercise and amusement. Their decorations v/ere most splen- 
did ; indeed, the exuberance of the ornaments alone was offen- 
sive to good taste. So overloaded with enrichments were the 
baths of Diocletian, that on an occasion of public festivity, 
great quantities of sculpture fell from the ceilings and entabla- 
tures, killing many of the people. 

213. — ^The three orders of Greece were introduced into 
Kome in all the richness and elegance of their perfection. 
But the luxurious Romans, not satisfied with the simple ele- 
gance of their refined proportions, sought to improve upon 
them by lavish displays of ornament. They transformed in 
many instances, the true elegance of the Grecian art into a 
gaudy splendour, better suited to their less refined taste. The 
Romans remodelled each of the orders : the Doric, {Fig. 126,) 
was modified by increasing the heignt of the column to 8 dia- 
meters ; by changing the echinus of the capital for an ovolo, 
or quarter round, and adding an astragal and neck below it ; 
by placing the centre, instead of one edge, of the first triglyph 



ARCHITECTURE. 



101 





Fig 127. 



102 AMERICAN HOUSE-CAEPENTER. 

over the centre of the column; and introducing horizontal 
instead of inclined mutules in the cornice, and in some instan 
ces dispensing with them altogether. The Ionic was modified 
bj diminishing the size of the volutes, and, in some specimens, 
introducing a new capital in which the volutes were diago- 
nally arranged, [Fig. 12 T.) This new capital has been termed 
modern Ionic. The favorite order at Home and her colonies 
was the Corinthian, {^Fig. 128.) But this order, the Roman 
artists in their search for novelty, subjected to many altera- 
tions — especially in the foliage of its capital. Into the upper 
part of this, they introduced the modified Ionic capital ; thus 
combining the two in one. This change was dignified with 
the importance of an order, and received the appellation, 
Composite, or Roman: the best specimen of wdiich is found in 
the Arch of Titus, (i^^p'. 129.) This style was not much used 
among the Romans themselves, and is but slightly appreciated 
now. 

214. — ^The Tuscan Oedek is said to have been introduced to 
the Romans by the Etruscan architects, and to have been the 
only style used in Italy before the introduction of the Grecian 
orders. However this may be, its similarity to the Doric 
order gives strong indications of its having been a rude imita- 
tion of that style : this is very probable, since history informs 
us that the Etruscans held intercourse with the Greeks at a 
remote period. The rudeness of this order prevented its ex- 
tensive use in Italy. All that is known concerning it is from 
Yitruvius — ^no remains of buildings in this style being found 
among ancient ruins. 

215. — For mills, factories, markets, barns^ stables, &c., where 
utility and strength are of more importance than beauty, the 
improved modification of this order, called the modem Tuscan, 
(Fig. 130,) will be useful ; and its simplicity recommends it 
where economy is desirable. 

216. — ^Egyptian Style. The architecture of the ancient 



AECHITEOTURE. 



103 



& 



• Sy OdyMUMJAMJ4iktJblJAa AJ^ 





Fig. 128. 



104 



AMEEICAlf HOUSE-CAKPENTEE. 




"52 ;4 

■■■■A 
t\'% 
35 '• 



59 I, 



?6;^ 



26^^ 



54,1 
37(4 

If 
.■*i;: 



LM-ftJl-0LO-O_flJ_fLJLlJ j 




l^ ^.J^.i^S^-^<\<<\<AS::.^^ViAU^b~^^^ 



w^ 




Tf IIT I K Ji: 



i!? tH m ITT gt BT T!V ITV— IT 



IIDQI 




ni.vrc',.5^ ^VN. 



Fir. 129, 



TUSCAIST ORDER. 



105 




106 ARCHITECTURE. 

Egyptians -to which that of the ancient Hindoos bears some re- 
semblance— is characterized by boldness of outline, solidity and 
grandeur. Tlie imazing labyrinths and extensive artificial lakes, 
the splendid jxilaces and gloomy cemeteries, the gigantic pyramids 
and towering obelisks, of the Egyptians, were works of immen- 
sity and durability ; and their extensive remains are enduring 
proofs of the enlightened skill of this once-powerful, but long since 
extinct nation. The principal features of the Egyptian Style of 
architecture are — uniformity of plan, never deviating from right 
lines and angles ; thick walls, having the outer surface slightly 
deviating inwardly from the perpendicular ; the whole building 
low ; roof flat, composed of stones reaching in one piece from pier 
to pier, these being supported by enormous columns, very stout in 
proportion to their height ; the shaft sometimes polygonal, having 
no base but with a great variety of handsome capitals, the foliage 
of these being of the palm, lotus and other leaves ; entablatures 
having simply an architrave, crowned with a huge cavetto orna- 
mented with sculpture ; and the intercolumniation very narrow, 
usually li diameters and seldom exceeding 2^. In the remains 
of a temple, the walls w^ere found to be 24 feet thick ; and at the 
gates of Thebes, the walls at the foundation were 50 feet thick 
and perfectly solid. The immense stones of which these, as well 
as Egyptian walls generally, were built, had both their inside and 
outside surfaces faced, and the joints throughout the body of the 
wall as perfectly close as upon the outer surface. For this reason, 
as well as that the buildings generally partake of the pyramidal 
form, arise their great solidity and durability. The dimensions 
and extent of the buildings may be judged from the temple ot 
Jupiter at Thebes, which was 1400 feet long and 300 feet wide — 
exclusive of the porticos, of which there was a great number. 

It is estimated by Mr. Gliddon, U. S. consul in Egypt, that not 
less than 25,000,000 tons of hewn stone wert employed in the 
erection of the Pyramids of Memphis alone, — or enough to con- 
struct 3,000 Bunker-Hill monuments. Some of the blocks are 40 



EGYPTIAN STYLE. 



1C7 



H. p. 




Fig. 131. 



108 ARCHITECTURE. 

feet long, and polished with emery to a surprising degree. It is 
conjectured that the stone for tliese pyramids was brought, by 
rafts and canals, from a distance of 6 or 7 hundred miles. 

217. — The general appearance of the Egyptian style of archi- 
tecture is that of solemn grandeur — amounting sometimes to 
sepulchral gloom. For this reason it is appropriate for cemete- 
ries, prisons, (fee. ; and being adopted for these purposes, it is 
gradually gaining favour. 

A great dissimilarity exists in the proportion, form and general 
features of Egyptian columns. In some instances, there is no 
uniformity even in those of the same building, each differing 
from the others either in its shaft or capital. For practical use 
in this country. Fig. 131 may be taken as a standard of this 
style. The Halls of Justice in Centre-street, New- York city, is 
a building in general accordance with the principles of Egyptian 
architecture. 

Buildings in General. 

218, — That style of architecture is to be preferred in whicii 
utility, stability and regularity, are gracefully blended with gran- 
deur and elegance. But as an arrangement designed for a warm 
country would be inappropriate for a colder climate, it would seem 
that the style of building ought to be modified to suit the wants 
of the people for whom it is designed. High roofs to resist the 
pressure of heavy snows, and arrangements for artificial heat, are 
indispensable in northern climes ; while they would be regarded 
as entirely out of place in buildings at the equator. 

219. — Among the Greeks, architecture was employed chiefly 
upon their temples and other large buildings ; and the proportions 
of the orders, as determined by them, when executed to such 
large dimensions, have the happiest effect. But when used for 
small buildingSjporticos, porches, <fec., especially in country-places, 
they are rather heavy and clumsy ; in such cases, more slendei 
proportions will be found to pr>duce a better effect. The 



AMERICAN HOUSE-Ci RPENTER. 109 

English cottage-style is rather more appropriate, and is becom- 
ing extensively practised for small buildings in the country. 

220. — Every building should bear an expression suited to its 
destination. If it be intended for national purposes, it should be 
magnificent — grand ; for a private residence, neat and modest ^ 
for a banqueting-house, gay and splendid ; for a monument or 
cemetery, gloomy — melancholy ; or, if for a church, majestic and 
graceful. By some it has been said — "somewhat dark and 
gloomy, as being favourable to a devotional state of feeling ;" but 
such impressions can only result from a misapprehension of the 
nature of true devotion. "Her ways are ways of pleasantness', 
and all her paths are peace." The church should rather be a type 
of that brighter world to which it leads. 

221. — However happily the several parts of an edifice may bo 
disposed, and however pleasing it may appear as a whole, yet 
much depends upon its site^ as also upon the character and style 
of the structures in its immediate vicinity, and the degree of cul- 
tivation of the adjacent country. A splendid country-seat should 
have the out-houses and fences in the same style with itself, the 
trees and shrubbery neatly trimmed, and the grounds well cul- 
tivated. 

222. — Europeans express surprise that so many houses in this 
country are built of wood. And yet, in a new country, where 
wood is plenty, that this should be so is no cause for wonder. 
Still, the practice should not be encouraged. Buildings erected 
with brick or stone are far preferable to those of wood ; they are 
more durable ; not so liable to injury by fire, nor to need repairs : 
and will be found in the end quite as economical. A wooden 
house is suitable for a temporary residence only ; and those who 
would bequeath a dwelling to their children, will endeavour to 
buil'ii with a more durable material. Wooden cornices and gut- 
ters, attached to brick houses, are objectionable- -not only on ac- 
count of their frail nature, but also because they render the build- 
ing liable to destruction by fire. 



no 



AMERICAN HOIJSE-CARPENTEE. 




AECHITECTURE. Ill 

223. — Dwelling houses are built of various dimensions and 
styles, according to their destination ; and to give designs and 
directions for their erection, it is necessary to know their situa- 
tion and object. A dwelling intended for a gardener, would 
require very different dimensions and arrangements from one 
intended for a retired gentleman — with his servants, horses, 
&c. ; nor would a house designed for the city be appropriate 
for the country. For city houses, arrangements that would be 
convenient for one family might be very inconvenient for two 
or more. Fig, 132, 133, 134, 135, 136, and 137, represent the 
ichnogrwphiGal jprojection^ or ground-plan, of the floors of an 
ordinary city house, designed to be occupied by one family 
only. Fig. 139 is an elevation^ or front-view, of the same 
house : all these plans are drawn at the same scale — which is 
that at the bottom of Fig. 139. 

Fig. 132 is a Plan of the TJnder-Cellar. 

a^ is the coal-vault, 6 by 10 feet. 

J, is the furnace for heating the house. 

c, d^ are front and rear areas. 

Fig. 133 is a Plan of the Basement. 

a., is the library, or ordinary dining-room, 15 by 20 feet. 
5, is the kitchen, 15 by 22 feet. 
<?, is the store-room, 6 by 9 feet, 
(f, is the pantry, 4 by Y feet. 
e^ is the china closet, 4 by 7 feet, 
y, is the servants' water-closet. 
g^ is a closet. 

A, is a closet with a dumb-waiter to the first story above. 
^, is an ash closet under the front stoop. 
^', is the kitchen-range. 

X', is the sink for washing and drawing water. 
?, are wash trays. 



112 



AMEEICAN HOUSE-CAKPENTEE. 




Fig. 135. 

Second Story. 



AECHITECTUKE. 113 

Fig. 134 is a Plan of the Birst Story. 

«, is the parlor, 15 by 34 feet. 

J, is the dining-room, 16 by 23 feet. 

c, is the vestibule. 

^, is the closet containing the dumb-waiter from the basement. 

/*, is the closet containing butler's sink. 

^, ^, are closets. 

A, is a closet for hats and cloaks. 

^,y, are front and rear balconies. 

Fig. 135 is the Second Story. 

«, <z, are chambers, 15 by 19 feet. 

5, is a bed-room, 7^ by 13 feet. 

c, is the bath-room, T^ by 13 feet. 
€/, d^ are dressing-rooms, 6 by 7i feet. 

6, 6, are closets. 
y,y, are wardrobes. 

^, ^, are cupboards. 

i^^'^. 136 is the Third Story. 

^, ^, are chambers, 15 by 19 feet. 
J, 5, are bed-rooms, Y5 by 13 feet. . 
c, c, are closets. 

e?, is a linen closet, 5 by 7 feet. 
6, ^, are dressing-closets. 
f^f^ are wardrobes. 
^, ^, are cupboards. 

Fig. 137 is the Fourth Story. 

<^, «^, are chambers, 14 by 17 feet. 
5, 5, are bed-rooms, 8 J by 17 feet. 
c, c, c, are closets. 
d^ is the step-ladder to the root. 

15 



114 



AMERIOAN HOUSE-CARPENTER. 




Fig. 137. 
Fourth Story. 



AKCniTECTURE. 115 

Fig. 13S is the Section of the House sho^A•ing the heights of 
the several stories. 

Fig. 139 is the Front Elevation. 

The size of the house is 25 feet front by 55 feet deep ; this 
is about the average depth, although some are extended to 60 
and 65 feet in depth. 

These are introduced to give some general ideas of the prin- 
ciples to be folloAved in designing city houses. In placing the 
chimneys in the parlours, set the chimney-breasts equi-distant 
from the ends of the room. The basement chimney-breasts 
may be placed nearly in the middle of the side of the room, as 
there is but one flue to pass through the chimney-breast above ; 
but in the second story, as there are two flues, one from the 
basement and one from the parlour, the breast will have to be 
placed nearly perj)endicular over the j^arlour breast, so as to 
receive the flues ^vithin the jambs of the fire-place. As it is 
desirable to have the chimney-breast as near the middle of the 
room as possible, it may be placed a few inches towards that 
point from over the breast below. So in arranging those of 
the stories above, always make provision for the flues from 
below. 

224. — In placing the stairs, there should be at least as much 
room in the passage at the side of the stairs, as upon them ; 
and in regard to the length of the passage in the second story, 
there must be room for the doors which open from each of the 
principal rooms into the hall, and more if the stairs require it. 
Having assigned a position for the stairs of the second story, 
now generally placed in the centre of the depth of the house, 
let the winders of the other stories be placed perpendicularly 
over and under them ; and be careful to provide for head- 
room. To ascertain this, when it is doubtful, it is well to draw 
a vertical section of the whole stairs ; but in ordinary cases, 
this is not necessary. To dispose the windows properly, the 



116 



AMERICAN HOrSE-CAEPENTEE. 




Fig. 139. 
iflevaticn. 



ARCniTECTUEE. 117 

middle window of each story should be exactly in the middle 
of the front; but the pier between the two windows which 
light the parlour, should be in the centre of that room ; be- 
cause when chandeliers or any similar ornaments, hang from 
the centre-pieces of the parlour ceilings, it is important, in 
order to give the better effect, that the pier-glasses at the 
front and rear, be in a range with them. If both these ob- 
jects cannot be attained, an approximation to each must be 
attempted. The piers should in no case be less in width than 
the window openings, else the blinds or shutters when thrown 
open will interfere with one anotlier ; in general practice, it is 
well to make the outside piers § of the width of one of the 
middle piers. When this is desirable, deduct the amount of 
the three openings from the width of the front, and the re- 
mainder will be the amount of the width of all the piers ; 
divide this by 10, and the product wdll be § of a middle pier ; 
and then, if the parlour arrangements do not interfere, give 
tv^'ice this amount to each corner pier, and three times the 
same amount to each of the middle piers. 

PRINCIPLES OF AKCHITECTURE. 

225. — In the construction of the first habitations of men, 
frail and rude as they must have been, the first and principal 
object was, doubtless, utility — a mere shelter from sun and 
rain. But as successive storms shattered the poor tenement, 
man was taught by experience the necessit}^ of building with 
an idea to durabilit}^ And when in his walks abroad, the 
symmetry, proportion and beauty of nature met his admiring 
gaze, contrasting .so strangely with the misshapen and dispro- 
portioned work of his own hands, he was led to make gradual 
changes ; till his abode was rendered not only commodious 
and durable, but pleasant in its appearance ; and building 
became a fine art, having utility for its basis. 



118 A^EERICAN HOUSE-CAEPENTEK. 

226. — In all designs for buildings of importance, utility, 
durability and beauty, the first great principles of architec- 
ture, should be pre-eminent. In order tliat the edifice be 
useful, commodious and comfortable, the arrangement of the 
apartments should be such as to fit them for their several des- 
tinations ; for public assemblies, oratory, state, visitors, retir- 
ing, eating, reading, sleeping, bathing, dressing, &c. — these 
should each have its own peculiar form and situation. To 
accomplish this, and at the same time to make their relative 
situation agreeable and pleasant, producing regularity and 
harmony, require in some instances much skill and sound 
judgment. Convenience and regularity are very important, 
and each should have due attention ; yet when both cannot 
be obtained, the latter should in most cases give place to the 
former. A building that is neitlier convenient nor regular, 
whatever other good qualities it may possess, will be sure of 
disapprobation. 

227. — The utmost importance should be attached to such 
arrangements as are calculated to promote health : among 
these, ventilation is by no means the least. For this purpose, 
the ceilirgs of the apartments should have a respectable 
height ; and the sky-light, or any part of the roof that can be 
made moveable, should be arranged with cord and pullies, so 
as to be easily raised and lowered. Small openings near the 
ceiling, that may be closed at pleasure, should be made in the 
partitions that separate the rooms from the passages — espe- 
cially for those rooms which are used for sleeping apartments. 
All the apartments should be so arranged as to secure their 
being easily kept dry and clean. In dwellings, suitable apart- 
ments should be fitted up for hathing with- all the necessary 
apparatus for conveying the water. 

22^. — To i sure stability in an edifice, it should be designed 
upon well-known geometrical principles : such as science has de- 
monstrated to be necessary and sufficient fir firmness and dura- 



AMERICAN HOUSIi-CARPENTER. 119 

bility. It is well, also, that it have the appearance of stability as 
well as the reality ; for should it seem tottering and unsafe, the 
sensation of fear, rather than those of admiration and pleasure, 
will be excited in the beholder. To secure certainty and accu- 
racy in the application of those principles, a knowledge of the 
strength and other properties of the materials used, is indispensa- 
ble ; and in order that the whole design be so made as to be 
capable of execution, a practical knowledge of the requisite 
mechanical operations is quite important. 

229. — The elegance of an architectural design, although chiefly 
depending upon a just proportion and harmony of the parts, will 
be promoted by the introduction of ornaments — provided this be 
judiciously performed. For enrichments should not only be of a 
proper character to suit the style of the building, but should also 
have their true position, and be bestowed in proper quantity. The 
most common fault, and one which is prominent in Roman archi- 
tecture, is an excess of enrichment : an error which is carefully 
to be guarded against. But those who take the Grecian models 
for their standard, will not be liable to go to that extreme. In 
ornamenting a cornice, or any other assemblage of mouldings, at 
least every alternate member should be left plain ; and those that 
are near the eye should be more finished than those which are dis- 
tant. Although the characteristics of good architecture are utili- 
ty and elegance, in connection with durability, yet some buildings 
are designed expressly for use, and others again for ornament : in 
the former, utility, and in the latter, beauty, should be the gov- 
erning principle. 

230. — The builder should be intimately acquainted with the 
principles upon which the essential, elementary parts of a build- 
ing are founded. A scientific knowledge of these will insure 
certainty and security, and enable the mechanic to erect the most 
extensive and lofty edifices with confidence. The more important 
parts are the foundation, the column, the wall,' the lintel, the arch, 
ihe vault, the dome and the roof. A separate description of the 



120 ARCHITECTURE. 

peculiarities of each, would seem to be necessary ; and cannol 
perhaps be better expressed than in the following language of a 
modern writer on this subject. 

231. — "In laying the Foundation of any building, it is ne- 
cessary to dig to a certain depth in the earth, to secure a solid 
basis, below the reach of frost and common accidents. The 
most solid basis is rock, or gravel which has not been moved. 
Next to these are clay and sand, provided no other excavations 
have been made in the immediate neighbourhood. From this 
basis a stone wall is carried up to the surface of the ground, and 
constitutes the foundation. Where it is intended that the super- 
structure shall press unequally, as at its piers, chimneys, or 
columns, it is sometimes of use to occupy the space between the 
points of pressure by an inverted arch. This distributes the 
pressure equally, and prevents the foundation from springing be- 
tween the different points. In loose or muddy situations, it is 
always unsafe to build, unless we can reach the solid bottom 
below. In salt marshes and flats, this is done by depositing tim- 
bers, or driving wooden piles into the earth, and raising walls 
upon them. The preservative quality of the salt will keep these 
timbers unimpaired for a great length of time, and makes the 
foundation equally secure with one of brick or stone. 

238. — The simplest member in any building, though by no 
means an essential one to all, is the Column, ot pillar. This is 
a perpendicular part, commonly of equal breadth and thickness, 
not intended for the purpose of enclosure, but simply for the sup- 
port of some part of the superstructure. The principal force 
which a column has to resist, is that of perpendicular pressure. 
In its shape, the shaft of a column should not be exactly cylin- 
drical, but, since the lower part must support the weight of the 
superior part, in addition to the weight which presses equally on 
the vvhole column, the thickness should gradually decrease from 
bottom to top. The outline of columns should be a little curved, 
so as to represent a portion of a very long spheroid, or paraboloid, 



AMERICAN HOUSE-CARPENTER. 121 

rather than of a cone. This figure is the joint result of two cal- 
culations, independent of beauty of appearance. One of these 
is, that the form best adapted for stability of base is that of a 
cone; the other is, that the figure, which would be of equal 
strength throughout for supporting a superincumbent weight, 
would be generated by the revolution of two parabolas round the 
axis of the column, the vertices of the curves being at its ex- 
tremities. The swell of the shafts of columns was called the ci- 
tasis by the ancients. It has been lately found, that the columns 
of the Parthenon, at Athens, which have been commonly sup. 
posed straight, deviate about an inch from a straight line, and 
that their greatest swell is at about one third of their height. 
Columns in the antique orders are usually made to diminish one 
sixth or one seventh of their diameter, and sometimes even one 
fourth. The Gothic pillar is commonly of equal thickness 
throughout. 

233. — The Wall, another elementary part of a building, may 
be considered as the lateral continuation of the column, answer- 
ing the purpose both of enclosure and support. A wall must 
diminish as it rises, for the same reasons, and in the same propor- 
tion, as the column. It must diminish still more rapidly if it ex- 
tends through several stories, supporting weights at difierent 
heights. A wall, to possess the greatest strength, must also con- 
sist of pieces, the upper and lower surfaces of which are horizon- 
tal and regular, not rounded nor oblique. The walls of most of 
the ancient structures which have stood to the present time, are 
constructed in this manner, and frequently have their stones bound 
together with bolts and cramps of iron. The same method is 
adopted in such modern structures as are intended to possess great 
strength and durability, and, in some cases, the stones are even 
dove-tailed together, as in the light-houses at Eddystone and Bell 
Rock, But many of our modern stone walls, for the sake of 
cheapness, have only one face of the stones squared, the inner 

half of the wall being completed with brick ; so that they can, 

16 



122 



ARCHITECTURE. 



m reality, be considered only as brick walls faced with stone 
Such walls are said to be liable to become convex outwardly, from 
the difference in the shrinking of the cement. Rubble walls are 
made of rough, irregular stones, laid in mortar. The stones 
should be broken, if possible, so as to produce horizontal surfaces 
The coffer walls of the ancient Romans were made by enclosing 
successive portions of the intended wall in a box, and filling it 
with stones, sand, and mortar, promiscuously. This kind of 
structure must have been extremely insecure. The Pantheon, 
and various other Roman buildings, are surrounded with a double 
brick wall, having its vacancy filled up with loose bricks and 
cement. The whole has gradually consolidated into a mass ot 
great firmness. 

The reticulated walls of the Romans, having bricks with 
oblique surfaces, would, at the present day, be thought highly 
unphilosophical. Indeed, they could not long have stood, had it 
not been for the great strength of their cement. Modern brick 
walls are laid with great precision, and depend for firmness more 
upon their position than upon the strength of their cement. The 
bricks being laid in horizontal courses, and continually overlaying 
each other, or breaking joints, the whole mass is strongly inter- 
woven, and bound together. Wooden walls, composed of timbers 
covered with boards, are a common, but more perishable kind. 
They require to be constantly covered with a coating of a foreign 
substance, as paint or plaster, to preserve them from spontaneous 
decomposition. In some parts of France, and elsewhere, a kind 
of wall is made of earth, rendered compact by ramming it in 
moulds or cases. This method is called building in pise, and is 
much more durable than the nature of the material would lead 
us to suppose. Walls of all kinds are greatly strengthened by 
angles and curves, also by projections, such as pilasters, chimneys 
and buttresses. These projections serve to increase the breadth 
of the foundation, and are always to be made use of in large 
buildings, and in walls of considerable length. 



AMERICAN HOUSE-CARPENTER. 



123 



234.— The Lintel, or 6ea??i, extends in a right line over e, 
vacant space, from one column or wall to another. The strength 
of the lintel will be greater in proportion as its transverse vertical 
diameter exceeds the horizontal, the strength being always as the 
square of the depth. The Jloor is the lateral continuation or 
connection of beams by means of a covering of boards. 

235. — The Arch is a transverse member of a building, an- 
swering the same purpose as the lintel, but vastly exceeding it in 
strength. The arch, unlike the lintel, may consist of any num- 
ber of constituent pieces, without impairing its strength. It is, 
however, necessary that all the pieces should possess a uniform 
shape, — the shape of a portion of a wedge, — and that the joints, 
formed by the contact of their surfaces, should pomt towards a 
common centre. In this case, no one portion of the arch can be 
displaced or forced inward ; and the arch cannot be broken by 
any force which is not sufficient to crush the materials of which 
it is made. In arches made of common bricks, the sides of which 
are parallel, any one of the bricks might be forced inward, were 
it not for the adhesion of the cement. Any two of the bricks, 
however, by the disposition of their mortar, cannot collective- 
ly be forced inward. An arch of the proper form, when com- 
plete, is rendered stronger, instead of weaker, by the pressure of 
a considerable weight, provided this pressure be uniform. While 
building, however, it requires to be supported by a centring of 
the shape of its internal surface, until it is complete. The upper 
stone of an arch is called the key-stone, but is not more essential 
than any other. In regard to the shape of the arch,, its most 
simple form is that of the semi-circle. It is, however, very fre- 
quently a smaller arc of a circle, and, still more frequently, a por- 
tion of an ellipse. The simplest theory of an arch supporting 
itself only, is that of Dr. Hooke. The arch, when it has only 
its own weight to bear, may be considered as the inversion of a 
chain, suspended at each end. The chain hangs in such a form, 
that the weight of each link or portion is held in equilibrium by 



124: ARCHITECTURE. 

the result of two forces acting at its extremities ; and these forces, 
or tensions, are produced, the one by the weight of the portion of 
the chain below the link, the other by the same weight increased 
by that of the link itself, both of them acting originally in a ver- 
tical direction. Now, supposing the chain inverted, so as to con- 
stitute an arch of the same form and weight, the relative situa- 
tions of the forces will be the same, only they will act in contrary 
directions, so that they are compounded in a similar manner, and 
balance each other on the same conditions. 

The arch thus formed is denominated a catenary arch. In 
common cases, it differs but little from a circular arch of the extent 
of about one third of a whole circle, and rising from the abut- 
ments with an obliquity of about 30 degrees from a perpendicu- 
lar. But though the catenary arch is the best form for support- 
ing its own weight, and also all additional weight which presses 
in a vertical direction, it is not the best form to resist lateral 
pressure, or pressure like that of fluids, acting equally in all direc- 
tions. Thus the arches of bridges and similar structures, when 
covered with loose stones and earth, are pressed sideways, as well 
as vertically, in the same manner as if they supported a weight 
of fluid. In this case, it is necessary that the arch should arise 
more perpendicularly from the abutment, and that its general 
figure should be that of the longitudinal segment of an ellipse. 
In small arches, in common buildings, where the disturbing 
force is not great, it is of little consequence what is the shape of 
the curve. The outlines may even be perfectly straight, as in tht 
tier of bricks which we frequently see over a window. This is, 
strictly speaking, a real arch, provided the surfaces of the bricks 
tend tovv^ards a common centre. It is the v/eakest kind of arch, 
and a part of it is necessarily superfluous, since no greater portion 
can act in supporting a weight above it, than can be included be- 
tween two curved or arched lines. 

Besides the arches already mentioned, various others are in use. 
The acute or lancet arch, much used in Gothic architecture, is 



AMERICAN HOUSE-CARPENTER. 125 

described usually from two centres outside the arch. It is a 
strong arch for supporting vertical pressure. The rampant arch 
is one in which the two ends spring from unequal heights. The 
horseshoe or Moorish arch is described from one or more centres 
placed above the base line. In this arch, the lower parts are in 
danger of being forced inward. The ogee arch is concavo-con- 
vex, and therefore fit only for ornament. In describing arches^ 
the upper surface is called the extrados^ and the inner, the in- 
trados. The springing lines are those where the intrados meets 
the abutments, or supporting walls. The span is the distance 
from one springing line to the other. The wedge-shaped stones, 
which form an arch, are sometimes called voussoirsj the upper- 
most being the key-stone. The part of a pier from which an 
arch springs is called the impost, and the curve formed by the 
upper side of the voussoirs, the archivolt. It is necessary that 
the walls, abutments and piers, on which arches are supported, 
should be so firm as to resist the lateral thrust^ as well as vertical 
pressure, of the arch. It will at once be seen, that the lateral or 
sideway pressure of an arch is very considerable, when we recol- 
lect that every stone, or portion of the arch, is a wedge, a part of 
whose force acts to separate the abutments. For want of atten- 
tion to this circumstance, important mistakes have been committed, 
the strength of buildings materially impaired, and their ruin ac- 
celerated. In some cases, the want of lateral firmness in the 
walls is compensated by a bar of iron stretched across the span of 
the arch, and connecting the abutments, like the tie-beam of a 
roof. This is the case in the cathedral of Milan and some other 
Gothic buildings. 

In an arcade, or continuation of arches, it is only necessary that 
the outer supports of the terminal arches should be strong enough 
to resist horizontal pressure. In the intermediate arches, the lat- 
eral force of each arch is counteracted by the opposing lateral 
force of the one contiguous to it. In bridges, however, where 
individual arches are liable to be destroyed by accident, it is desi 



126 ARCHITECTURE. 

rable that each of the piers should possess sufficient horizontal 
strength to resist the lateral pressure of the adjoining arches. 

236. — The Vault is the lateral continuation of an arch, serving 
to cover an area or passage, and bearing the same relation to the 
arch that the wall does to the column. A simple vault is con- 
structed on the principles of the arch, and distributes its pressure 
equally along the walls or abutments. A complex or groined 
vault is made by two vaults intersecting each other, in which 
case the pressure is thrown upon springing points, and is greatly 
increased at those points. The groined vault is common m 
Gothic architecture. 

237. — The Dome, sometimes called cupola, is a concave cover- 
ing to a building, or part of it, and may be either a segment of a 
sphere, of a spheroid, or of any similar figure. When built of 
stone, it is a very strong kind of structure, even more so than the 
arch, since the tendency of each part to fall is counteracted, not 
only by those above and below it, but also by those on each side. 
It is only necessary that the constituent pieces should have a 
common form, and that this form should be somewhat like the 
frustum of a pyramid, so that, when placed in its situation, its 
four angles may point toward the centre, or axis, of the dome. 
During the erection of a dome, it is not necessary that it should 
be supported by a centring, until complete, as is done in the arch. 
Each circle of stones, when laid, is capable of supporting itself 
without aid from those above it. It follows that the dome may 
be left open at top, without a key-stone, and yet be perfectly 
secure in this respect, being the reverse of the arch. The dome 
of the Pantheon, at Rome, has been always open at top, and yet 
has stood unimpaired for nearly 2000 years. The upper circle 
of stones, though apparently the weakest, is nevertheless often 
made to support the additional weight of a lantern or tower above 
it. In several of the largest cathedrals, there are two domes, one 
within the other, which contribute their joint support to the lan- 
tern, which rests upon the top. In these buildings, the dome 



AMERICAN HOUSE-CARPENTER. 



127 



rests upon a circular ^^'all, which is supported, in its turn, by 
arches upon massive pillars or piers. This construction is called 
building upon pendentives, and gives open space and room for 
passage beneath the dome. The remarks which have been made 
in regard to the abutments of the arch, apply equally to the walls 
immediately supporting a dome. They must be of sufficient 
thickness and solidity to resist the lateral pressure of the dome, 
which is very great. The walls of the Roman Pantheon are of 
great depth and solidity. In order that a dome in itself should be 
perfectly secure, its lower parts must not be too nearly vertical, 
since, in this case, they partake of the nature of perpendicular 
walls, and are acted upon by the spreading force of the parts above 
them. The dome of St. Paul's church, in London, and some 
others of similar construction, are bound with chains or hoops of 
iron, to prevent them from spreading at bottom. Domes which 
are made of wood depend, in part, for their strength, on their in- 
ternal carpentry. The Halle du Bled, in Paris, had originally a 
wooden dome more than 200 feet in diameter, and only one foot 
in thickness. This has since been replaced by a dome of iron 
(See Art. 389.) 

238. — The Roof is the most common and cheap method of 
covering buildings, to protect them from rain and other effects of 
the weather. It is sometimes flat, but more frequently oblique, in 
its shape. The flat or platform-rooi is the least advantageous for 
shedding rain, and is seldom used in northern countries. The 
pent roof, consisting of two oblique sides meeting at top, is the 
most common form. These roofs are made steepest in cold cli- 
mates, where they are liable to be loaded with snow. Where the 
four sides of the roof are all oblique, it is denominated a hipped 
roof, and where there aie two portions to the roof, of different ob- 
liquity, it is a curb, or mansard roof. In modern times, roofs 
are made almost exclusively of wood, though frequently covered 
with incombustible materials. The internal structure or carpen- 
try of roofs is a subject of considerable mechanical contrivance. 



128 ARCHITECTURE. 

The roof is supported by rafters, which abut on the walls on 
each side, like the extremities of an arch. If no other timbers 
existed, except the rafters, they would exert a strong lateral pres- 
.sure on the walls, tending to separate and overthrow them. To 
counteract this lateral force, a tie-beam, as it is called, extends 
across, receiving the ends of the rafters, and protecting the wall 
from their horizontal thrust. To prevent the tie-beam from 
sagging, or bending downward with its own weight, a king- 
post is erected from this beam, to the upper angle of the rafters, 
serving to connect the whole, and to suspend the weight of the 
beam. This is called trussing. Qneen-posts are sometimes 
added, parallel to the king-post, in large roofs ; also various other 
connecting timbers. In Gothic buildings, where the vaults do 
not admit of the use of a tie-beam, the rafters are prevented from 
spreading, as in an arch, by the strength of the buttresses. 

In comparing the lateral pressure of a high roof with that of a 
low one, the length of the tie-beam being the same, it will be 
seen that a high roof, from its containing most materials, may 
produce the greatest pressure, as far as weight is concerned. On 
the other hand, if the weight of both be equal, then the low roof 
will exert the greater pressure ; and this will increase in propor- 
tion to the distance of the point at which perpendiculars, drawn 
from the end of each rafter, would meet. In roofs, as well as in 
wooden domes and bridges, the materials are subjected to an in- 
ternal strain, to resist which, the cohesive strength of the material 
is relied on. On this account, beams should, when possible, be 
of one piece. Where this cannot be effected, two or more beams 
are connected together by splicing. Spliced beams are never so 
strong as wPiole ones, yet they may be made to approach the same 
strength, by affixing lateral pieces, or by making the ends overlay 
each other, and connecting them with bolts and straps of iron. 
The tendency to separate is also resisted, by letting the two pieces 
into each Ir^ther by the process called scarfing. Mori ices ^ in- 



AMERICAN HOUSE-CARPENTER. 129 

tended to truss or suspend one piece by another, should be formed 
upon similar principles. 

Roofs in the United States, after being boarded, receive a se- 
condary covering of shingles. When intended to be incombustible, 
they are covered with slates orearthern tiles, or with sheets of lead, 
cbpper or tinned iron. Slates are preferable to tiles, being lighter, 
and absorbing less moisture. Metallic sheets are chiefly used for 
flat roofs, wooden domes, and curved and angular surfaces, which 
require a flexible material to cover them, or have not a sufficient 
pitch to shed the rain from slates or shingles. Various artificial 
compositions are occasionally used to cover roofs, the most com- 
mon of which are mixtures of tar with lime, and sometimes with 
sand and gravel." — Ency. Am. (See Art, 354.) 

17 



SECTION III.— MOULDINGS, CORNICES, &c. 



MOULDINGS. 



239. — A moulding is so called, because of its being ol the 
same determinate shape along its whole length, as though the 
whole of it had been cast in the same mould or form. The regulai 
mouldings, as found in remains of ancient architecture, are eight 
in number ; and are known by the following names : 

J Annulet, band, cincture, fillet, listel or square. 



Fig. 140. 



Fig. 141. 



J Astragal or bead. 



) 



Torus or tore. 



Fig. 142. 



L 



Fig. 14a 



Scotia, trochilus or mouth. 



Fig. 144 



Ovolo, quarter-round or echinus. 



AMERICAN HOUSE-CARPENTER. 



131 




Cavetto, cove or hollow. 



Cymatium, or cyma-recta. 



Fig. 147. 



Inverted cymatiumj or cyma-reversa 



Ogee. 



Some of the terms are derived thus : fillet, from the French 
word//, thread. Astragal, from astragalos^ a bone of the heel 
— or the curvature of the heel. Bead, because this moulding, 
when properly carved, resembles a string of beads. Torus, or 
tore, the Greek for rope, which it resembles, when on the base ot 
a column. Scotia, from shotia, darkness, because of the strong 
shadow which its depth produces, and which is increased by the 
projection of the torus above it. Ovolo, from ovufn, an egg, 
which this member resembles, when carved, as in the Ionic capi- 
tal. Cavetto, from cavus, hollow. Cymatium, from kumaton 
a wave. 

240. — Neither of these mouldings is peculiar to any one of the 
orders of architecture, but each one is common to all ; and al- 
though each has its appropriate use, yet it is by no means con- 
fined to any certain position in an assemblage of mouldings 
The use of the fillet is to bind the parts, as also that of the astra- 
gal and torus, which resemble ropes. The ovolo and cyma-re- 
versa are strong at their upper extremities, and are therefore used 
to support projecting parts above them. The cyma-recta and 
cavetto, being weak at their upper extremities, are not used as 
supporters, but are placed uppermost to cover and shelter tht 
other parts. The scotia is introduced in the base of a column, to 



132 MOULDINGS, CORNICES, &C. 

separate the upper and lower torus, and to produce a pleasing 
variety and relief. The form of the bead, and that of the toms, 
is the same ; the reasons for giving distinct names to them are, 
that the torus, in every order, is always considerably larger than 
the bead, and is placed among the base mouldings, whereas the 
bead is never placed there, but on the capital or entablature ; the 
torus, also, is seldom carved, whereas the bead is ; and while the 
torus among the Greeks is frequently elliptical in its form, the 
bead retains its circular shape. While the scotia is the reverse of 
the torus, the cavetto is the reverse of the ovolo, and the cyma- 
recta and cyma-reversa are combinations of the ovolo and cavetto. 

241. — The curves of mouldings, in Roman architecture, were 
most generally composed of parts of circles ; while those of the 
Greeks were almost always elliptical, or of some one of the conic 
sections, but rarely circular, except in the case of the bead, which 
was always, among both Greeks and Romans, of the form of a 
semi-circle. Sections of the cone afford a greater variety of 
forms than those of the sphere ; and perhaps this is one reason 
why the Grecian architecture so much excels the Roman. The 
quick turnings of the ovolo and cyma-reversa, in particular, when 
exposed to a bright sun, cause those narrow, well-defined streaks 
of light, which give life and splendour to the whole. 

242. — K profile is an assemblage of essential parts and mould- 
ings. That profile produces the happiest effect which is com- 
posed of but few members, varied in form and size, and arranged 
so that the plane and the curved surfaces succeed each other al- 
ternately. 

243. — To describe tke Grecian torus and scotia. Join the 
extremities, a and 6, {Fig. 148;) and from/, the given projection 
oi the moulding, draw/ o, at right angles to the fillets ; from 6, 
draw h A, at right angles io a h ; bisect a b in c; join / and c, 
and upon c, with the radius, c/ describe the arc, / A, cutting b h 
in h ; through c, draw d e, parallel with the fillets ; make d c and 
c €j each equal to 6 A ; then d e and a b will be conjugate diame- 



AMERICAN HOUSE-CARPENTER. 



133 




Fig. 148. 



ters of the required ellipse. To describe the curve by intersec- 
tion of lines, proceed as directed at Art. 118 and note ; by a 
trammel, see Art. 116 ; and to find the foci, in order to describe it 
with a string, see Art. 115. 





d 




d 


'^ 




^^ 


< 




\ 


^ 






a 




h 




a 




Fig 


.149. 




Fig. 1' 


50. 



244. — Fig. 149 to 156 exhibit various modifications of the 
Grecian ovolo, sometimes called echinus. Fig. 149 to 153 are 



134 



MOULDINGS, CORNICES, &C. 





h- 1 


c 


^. 




I 



Fis:. 151. 




Fig. 152. 




? 




Fig. 158. 



Fig. 154. 







1 

a 


c, 










:^^^-^^^ 




^y^/ / 


77 




u 










Fig. 155. 



Fig. 156. 



elliptical, a h and 6 c being given tangents to the curve ; parallel 
to which, the semi-conjugate diameters, a d and d c, are drawn. 
In Fig. 149 and 150, the lines, a d and d c, are semi-axes, the 
tangents, a h and h c, being at right angles to each other. To 
draw the curve, see Art. 118. In Fig. 153, the curve is para- 
bolical, and is drawn according to Art. 127. In Fig, 155 and 156, 
the curve is hyperbolical, being described according to Art. 128. 
The length of the transverse axis, a b, being taken at pleasure 
in order to flatten the curve, a b should be made short in prc^r- 
tion to a c. 



AMERICAN HO^JSE-CARPENTER. 



135 





Fig. 158. 



Fig. 167. 



24:5. — To describe the Grecian cavetto^ {Fig;16^ and 158,) 
having the height and projection given, see Art. 118. 



m 



a 

/■ Ik 



Fig. 159. 



Fig. 160. 



246. — To describe the Grecian cyma-recta. When the pro- 
jection is more than the height, as at Fig. 159, make a b equal 
to the height, and divide abed into 4 equal parallelograms ; 
then proceed as directed in note to Art. 118. When the projec- 
tion is less than the height, draw d a, {Fig. 160,) at right angles 
to a b; complete the rectangle, abed; divide this into 4 equal 
rectangles, and proceed according to Art. 118. 



1 


1 




"^ 


^ 




a 


^\ 


m 


h^ 


^^ 



Fiff. 161 



Fig. 162. 



217. — To describe the Grecian cyma-reversa. Wlien the 



136 



MOULDINGS, CORNICES, &C. 



projection is more than the height, as at Fig. 161, proceed as di 
rected for the last figure ; the curve being the same as that, the 
positiononly being changed. When the projection is less than 
the height, draw a d, {Fig. 162,) at right angles to the fillet ; 
make a d equal to the projection of the moulding : then proceed 
as directed for Fig. 159. 

248. — Roman mouldings are composed of parts of circles, and 
have, therefore, less beauty of form than the Grecian. The bead 
and torus are of the form of the semi-circle, and the scotia, also, 
in some instances ; but the latter is often composed of two quad- 
rants, having difierent radii, as at Fig. 163 and 164, which re- 
semble the elliptical curve. The ovolo and cavetto are generally 
a quadrant, but often less. When they are less, as at Fig. 167, 
the centre is found thus : join the extremities, a and 6, and bisect 
ahin c ; from c, and at right angles to a b, draw c rf, cutting a 
level line drawn from a in d ; then d will be the centre. This 
moulding projects less than its height. When the projection is 
more than the height, as at Fig. 169, extend the line from c until 





Fig. 163. 



Fig. 164. 




Fig. 165. 



Fig. 166. 



AMERICAN HOUSE-CARPENTER. 



137 




Fig. 167. 




Tig. 168. 



a 



Fig. 169. 




Fig. 170. 



Fig. 171. 



1 

1 






J 




r 




1 









Fig. 172. 




Fig. 173. 




Fig. 174. 



18 



lU 



MOULDINGS, CORl^ICES, &C 






K. 


,^ 


2^ 


_J \ 





Fig 175. 



Fig. 176. 




rC jd 



Fig. 177. 



Fif.178. 



it cuts a perpendicular drawn from a, as at d ; and that will 
be the centre of the curve. In a similar manner, the centres 
are found for the mouldings at Fig, 164, 168, 170, US, 174, 
175, and 176. The centres for the curves at Fig, 177 and 178, 
are found thus : bisect the line, « 5, at c / upon a, o and 5, suc- 
cessively, with a G or oh for radius, describe arcs intersecting 
at d and d ; then those intersections will be the centres. 

249. — Fig. 179 to 186 represent mouldings of modern inven- 
tion. They have been quite extensively and successfully used 
in inside finishing. Fig. 179 is appropriate for a bed-moulding 
under a low projecting shelf, and is frequently used under man- 
tle-shelves. The tangent, i h^ is found thus : bisect the line, a h, 
at (?, and h c at d; from d, draw d e, at right angles to 6 5 / from 
5, draw hf, parallel toed; upon 5, with h dfor radius, describe 
the arc, df; divide this arc into 7 equal parts, and set one of the 
parts from 5, the limit of the projection, to o ; make o h equal to 
oe ; from A, through ^ draw the tangent, h i / divide hh,hc,ci 
and i a, each into a like number of equal parts, and draw the in- 



AMERICAN HOUSE-CARPENTER 




Fig. 179. 




Wig. 181. 



iiO MOULDINGS, CORNICES, &C. 





Fig. 182. 



Fig. 183. 




Fig 184. 



Fig. 185. 



Fig. 186. 



tersecting lines as directed at Art. 89. If a bolder form is 
desired, draw the tangent, i A, nearer horizontal, and describe 
an elliptic cnrve as shown in Fig. 148 and 181. Fig. 180 is 
much used on base, or skirting of rooms, and in deep panelling. 
The curve is found in the same manner as that of Fig. 179. In 
this case, however, where the moulding has so little projection 



AMERICAN HOUSE-CARPENTER. 



141 



in comparison with its height, the point, e, beir*g found as in the 
last figure, h s may be made equal to s e, instead of o e as in tne 
last figure. Fig. 181 is appropriate for a crown moulding of a 
cornice. In this figure the height and projection are given ; the 
direction of the diameter, a b, drawn through the middle of 
the diagonal, e /, is taken at pleasure ; and d cis parallel to a 
e. To find the length of d c, draw 6 h, at right angles to a 6 ; 
upon 0, with o f for radius, describe the arc,/ h, cutting b h in 
h ; then make o c and o d, each equal to b h* To draw the curve, 
see note to Art. 118. Fig. 182 to 186 are peculiarly distinct from 
ancient mouldings, being composed principally of straight lines ; 
the few curves they possess are quite short and quick 



H 


P. 






5 
4 
~2 

9 


15 




12| 


.y 


11 


/ 


10* 








10 







H.P. 


1 






U 
3i 

3i 

ii 

I 
1 

9 


15 


^ 


14i 






14i 

13 

Hi 


J 






7. 


lOi 




1 








10 







Fig. 187. 



Fig. 1S8. 



250.— Fi^. 187 and 188 are designs for antse caps. The 



♦ The manner of ascertaining the length of the conjugate diameter, <i c, in thia figure, 
and also in Fig. 148, 198 and 199 is new, and is important in this application. It is 
founded upon well-known mathematical principles, viz : All the parallelograms that may 
be circumscribed about an ellipsis are equal to one another, and consequently any one 
is equal to the rectangle of the two axes. And again : the sum of the squares of every 
pair of conjugate diameters is equal to the sum of the squares of the two axes. 



142 



AMEEICAK HOrSE-CAEPENTEB. 



diameter of the antse is divided into 20 equal parts, and the 
height and projection of the members, are regulated in accord- 
ance with those parts, as denoted under IT and: P^ height and 
projection. The projection is measured from the middle of 
tke antas. These will be found appropriate for porticos, door- 
ways, mantel-pieces, door and window trimmings, &c. The 
height of the antse for mantel-pieces, should be from 5 to 6 
diameters, having an entablature of from 2 to 2^ diameters. 
This is a good proportion, it being similar to the Doric order. 
But for a portico these proportions are much too heavy ; an 
antse, 15 diameters high, and an entablature of 3 diameters, 
will have a better appearance. 

COBNICES. 

251. — Fig. 189 to 197 are designs for eave cornices, and 
Fig. 198 and 199 are for stucco cornices for the inside finish 
of rooms. In some of these the projection of the uppermost 
member from the facia, is divided into twenty equal parts, 



7 



^^ 




Fig 189. 



MOULDINGS, COENICES* &C. 



U3 



and tlie various members are proportioned according to those 
parts, as figured under ^and P. 




Fig. 190. 



7" 



wMMm^mjjmmjMjmJMim?/ 






Fig. 191. 



14.4 



AMERICAN HOUSE-CAEPENTEE. 



r 



7 



Q 



J 



^ 



H27 



Fig.l9S. 




Fig. 193. 



MOULDINGS, COENICES, &C. 



145 




Fig. 194. 



H 


P. 








n 

5 

H 
If 

j 

8 

1 
25 


20 


1 




/ 


1 


J 








.' ^ « 




1 


3 











Fig. 195. 

19 



146 



AMERICAN HOUSE-CARPENTER. 



H. P. 



[UjSO 

- 



J4i 







H 


2i 


^ 



Fig. 196. 



:^ 




Fig. 197. 



MOULDINGS, CORNICES, dsC. 



147 




Fig. 198. 




Fig. 199. 



148 



AMERICAN HOUSE-CARPENTER. 




b 12 3 4c 
Fig. 200. 



252. — To proportion an eave cornice in accordance with the 
height of the building. Draw the line, a c, {Fig. 200,) and 
make b c and b a, each equal to 36 inches ; from b, draw b d, at 
right angles to a c, and equal in length to f of a c ; bisect b din 
e, and from a, through e, draw a f; upon a, with a c for radius, 
describe the arc, c/, and upon e, with e/for radius, describe the 
arc,/c?; divide the curve, df c, into 7 equal parts, as at 10, 20, 
30, <fcc., and from these points of division, draw lines to b c, pa- 
rallel tod b ; then the distance, b 1, is the projection of a cornice 
for a building 10 feet high ; b 2, the projection at 20 feet high ; 
b 3, the projection at 30 feet, &c. If the projection of a cornice for 
a building 34 feet high, is required, divide the arc between 30 and 
40 into 10 equal parts, and from the fourth point from 30, draw a 
line to the base, b c, parallel with b d ; then the distance of the 
point, at which that line cuts the base, from 6, will be the projec- 
tion required. So proceed for a cornice of any height within 70 
feet. The above is based on the supposition that 36 inches is the 
proper projection for a cornice 70 feet high. This, for general 
purposes, will be found correct ; still, the length of the line, b c, 
maybe varied to suit the judgment of those who think differ- 
ently. 

Having obtained the projection of a cornice, divide it into 20 
equal parts, and apportion the several members according to its 
destination — as is shown at Fig. 195, 196, and 197. 



MOULDINGS, CORNICES, &C. 

b 



149 




Fig. 201. 



Ji53. — To proportion a cornice according to a smaller given 

&ne. Let the cornice at Fig. 201 be the given one. Upon any 
point in the lowest hne of the lowest memberj as at a, with the 
height of the required cornice for radius, describe an intersecting 
arc across the uppermost line, as at b ; join a and b : then b 1 will 
be the perpendicular height of the upper fillet for the proposed cor- 
nice, 1 2 the height of the crown moulding — and so of all the 
members requiring to be enlarged to the sizes indicated on this 
line. For the projection of the proposed cornice, draw a d, at right 
angles to a 6, and c d, at right angles to be; parallel with c d, 
draw lines fj-om each projection of the given cornice to the line, 
ad; then e d will be the required projection for the proposed 
cornice, and the perpendicular lines falling upon e d will indicate 
the proper projection for the members. 

254. — To projwrtion a cornice according to a larger given 
one. Let J., {Fig. 202,) be the given cornice. Extend a o to b, 
and draw c d, at right angles to ab; extend the horizontal lines 
of the cornice. A, until they touch o d ; place the height of the 
proposed cornice from o to e, and join / and e ; upon o, Avith the 
projection of the given cornice, o a, for radius, describe the quad- 
rant, ad ; from d, draw d 6, parallel to/ e ; upon o, with o b for 
radius, describe the quadrant, be; then o c will be the proper pro- 
jection for the proposed cornice. Join a and c ; draw lines from the 



150 



AMERICAN HOUSE-CARPENTER. 



S'yy''^ ^^ 


c . 




1 
1 


1^- 










I / 


/ / 


/ / 








E//// 






A 


/ 


// / 




A 


f / 






V^ 


^ / 



Fig. 202. d 



projection of the different members of the given cornice to a a^ 
parallel to o d ; from these divisions on the line, a o, draAV lines 
to the line, o c, parallel to a c ; from the divisions on the line, of. 
draw lines to the line, o e, parallel to the line, f e ; then the di- 
visions on the lines, o e and o c, will indicate the proper height and 
projection for the different members of the proposed cornice. In 
this process, we have assumed the height, o e, of the proposed 
cornice to be given ; but if the projection, o c, alone be given, we 
can obtain the same result by a different process. Thus : upon o, 
with c for radius, describe the quadrant, ch ; upon o, with o a 
for radius, describe the quadrant, ad ; join d and h ; from/, draw 
f e. parallel to d b ; then o e will be the proper height for the pro- 
posed cornice, and the height and projection of the different mem- 
bers can be obtained by the above directions. By this problem, 
a cornice can be proportioned according to a smaller given one 
as well as to a larger ; but the method described in the previous 
article is much more simple for that purpose. 

255. — To find the angle-bracket for a cornice. Let A, {Fig. 
203,) be the wall of the building, and B the given bracket, which, 
for the present purpose, is turned down horizontally. The angle- 
bracket, C, is obtained thus : through the extremity, a, and paral- 



MOULDINGS, CORNICESj &C. 



151 





A 


G 




\ 


(■ 


.,.___, 




^■/ 


^^y\ ^ 




n 


■ ^' 




d^ 


.\^ 




^ 


3 3\ 



g Fig. 203. 




Fig. 204. 



lei with the wall, / d^ draw the line, ah ; make e c equal a /, 
and through c, draw c &, parallel with e d ; join c? and 5, and from 
the several angular points in 5, draw ordinates to cut d 6 in 1, 2 
and 3 ; at those points erect lines perpendicular to d b ; from h, 
draw h gj parallel to/ a ; take the ordinates, 1 o, 2 o, (fee, at B, 
and transfer them to C, and the angle-bracket, C, will be defined. 
In the same manner, the angle-bracket for an internal cornice, or 
the angle-rib of a coved ceiling, or of groins, as at Fig. 204, can 
be found. 

256. — A level crown moulding being given, to find the raking 
Wjoulding and a level return at the top. Let A, {Fig. 205,) be 
the given moulding, and A b the rake of the roof. Divide the 
curve of the given moulding into any number of parts, equal or 
unequal, as at 1, 2, and 3 ; from these points, draw horizontal 
lines to a perpendicular erected from c ; at any convenient place 
on the rake, as at B, draw a c, at right angles to Ah ; also, from 
6, draw the horizontal line, b a ; place the thickness, d a, of the 
moulding at A, from b to a, and from a, draw the perpendicular 
line, a e ; from the points, 1, 2, 3, at A, draw lines to C, parallel 
io Ah ; make a 1, a 2 and a 3, at i3 and at C, equal to a 1, (fee, 
at A ; through the points, 1, 2 and 3, at B, trace the curve — this 
will be the proper form for the raking moulding. From 1, 2 and 



152 



AMERICAN HOUSE-CARPRNTER. 




Fig 205. 



3, at Cj drop perpendiculars to the corresponding ordinates from 
1. 2 and 3, at J. /through the poirts of intersection, trace the 
curve — this will be the proper form lor the return at the top. 



SECTION IV.— FRAMING. 



257. — T)iis subject is, to the carpenter, of the highest impor- 
tance ; and deserves more attention and a larger place in a volume 
of this kind, than is generally allotted to it. Something, indeed, 
has been said upon the geometrical principles, by which the seve- 
ral lines for the joints and the lengths of timber, may be ascer- 
tained ; yet, besides this, there is much to be learned. For how- 
ever precise or workmanlike the joints may be made, what will 
it avail, should the system of framing, from an erroneous position 
of its timbers, (fee, change its form, or become incapable of sus- 
taining even its own weight ? Hence the necessity for a know- 
ledge of the laws of pressure and the strength of timber. These 
bemg once understood, we can with confidence determine the best 
position and dimensions for the several timbers which compose a 
floor or a roof, a partition or a bridge. As systems of framing 
are more or less exposed to heavy weights and strains, and, in 
case of failure, cause not only a loss of labour and material, but 
frequently that of life itself, it is very important that the materials 
employed be of the proper quantity and quality to serve their des 
tination. And, on the other hand, any superfluous material is not 
only useless, but a positive injury, it being an unnecessary load 
upon the points of support. It is necessary, therefore, to know 

20 



154: AMERICAN HOUSE-CARPENTER. 

the least quantity of timber that will suffice for strength. The 
greatest fault in framing is that of using an excess of materied. 
Economy, at least, would seem to require that this evil be abated. 

Before proceeding to considei the principles upon which a sys- 
tem of framing should be constructed, let us attend to a few of 
the elementary laws in Mechanics, which will be found to be of 
great value in determining those principles. 

258. — Laws of Pressure. (1.) A heavy body always 
exerts a pressure equal to its own weight in a vertical direction. 
Example: Suppose an iron ball, weighing iOO lbs., be supported 
upon the top of a perpendicular post, [Fig. 220 ;) then the 
pressure exerted upon that post will be equal to the weight of the 
oall; viz., 100 lbs. (2.) But if two inclined posts, (Fi^. 206,) 
be substituted for the perpendicular support, the united pressures 
upon these posts will be more than equal to the weight, and will 
be in proportion to their position. The farther apart their feet are 
spread the gr-eater will be the pressure, and vice versa. Hence 
tremendous strains may be exerted by a compciratively small 
weight. And it follows, therefore, that a piece of timber intend- 
ed for a strut or post, should be so placed that its axis may coin- 
cide, as near as possible, with the direction of the pressure. The 
direction of the pressure of the weight, TF, {Fig. 206^) is in the 
vertical line, h d ; and the weight, TF, would fall in that line, if 
the two posts were removed, hence the best position for a support 



w 




T\z. 206. 



FRAMING. 155 

for the weight would be in that line. But, as it rarely occurs 
in systems of framing that weights can be supported by any 
single resistance, they requiring generally two or more sup- 
ports, (as in the case of a roof supported by its rafters,) it be- 
comes important, therefore, to know the exact amount of pres- 
sure any certain weight is capable of exerting upon oblique 
supports. Now it has been ascertained that the three Hues of 
a triangle, drawn parallel with the direction of three concur- 
ring forces in equilibrium, are in proportion respectively to 
these forces. For example, in Fig. 206, we have a represen- 
tation of three forces concurring in a point, which forces are 
in equilibrium and at rest ; thus, the weight, TF", is one force, 
and the resistance exerted by the two pieces of timber are the 
other two forces. The direction in which the lirst force acts is 
vertical — downwards ; tKo dirp.ntion of the two other forces is 
in the axis of each piece of timber respectively. These three 
forces all tend towards the point, h. 

Draw the axes, a h and h c, of the two supports ; make h d 
vertical, and from d draw d e and d f parallel with the axes, 
5 G and h <z, respectively. Then the triangle, h d e^ has its 
lines parallel respectively with the direction of the three 
forces ; thus, 5 (^ is in the direction of the weight, W, d e 
parallel with the axis of the timber h c^ and ^ 5 is in the 
direction of the timber a h. In accordance with the principle 
above stated, the lengths of the sides of the triangle, h d e^ are 
in proportion respectively to the three forces aforesaid ; thus — . 

As the length of the line, h d^ 

Is to the number of pounds in the weight, TF", 

So is the length of the line, h 6, 

To the number of pounds' pressure resisted by the timber, 
a h. 
Again — 

As the length of the line, h d, 

Is to the number of pounds in the weight, W, 



156 AMERICAN HOUSE-CAEPENTER. 

So is the length of the line, d e^ 

To the number of pounds' pressure resisted by the timber, 
he. 
Audi again — 

As the length of the line, 5 6, 
Is to the pounds' pressure resisted by a 5, 
So is the length of the line, d e^ 
To the pounds' pressure resisted by h c. 
These proportions are more briefly stated thus— 
1st. hd : W::he:P, 

P being used as a symbol to represent the number of founds' 
pressure resisted by the timber, a I. 

'2nd. Id : W:: del Q, 

Q representing the number of pounds' pressure resisted by the 
timber, h c, 

Sd. heiFiideiQ. 

259. — ^This relation between lines and pressures is important, 
and is of extensive application in ascertaining the pressures 
induced by known weights throughout any system of framing. 
The parallelogram, h e df, is called the Parallelogram of 
Forces ; the two lines, h e and h /, being called the coTrvpo- 
nents^ and the line h d the resultant. Where it is required to 
find the components from a given resultant, {Fig. 206,) it is 
not needed to draw the fourth line, df for the triangle, h d e, 
gives the desired result. But when the resultant is to be 
ascertained from given components, {Fig. 212,) it is more con- 
venient to draw the fourth line. 

260. — The Resolution of Forces is the finding of two or 
more forces, which, acting in difi'erent directions, shall exactly 
balance the pressure of any given single force. To make a 
practical application of this, let it be required to ascertain 
the oblique pressure in Fig. 206. In this Fig. the line h d 
measures half an inch, (0-5 inch,) and the line h e three- 
tenths of an inch, (0*3 inch.) Now if the weight, TT, be sup- 



FRAMING. 



161 



posed to be 1200 pounds, then the first stated proportion 
above, 

hd : Wy.heiF, 
becomes 

0-5:1200 :: 0-3 : P. 
And since the product of the means divided by one of the 
extremes gives the other extreme, this proportion may be put 
in the form of an equation, thus — 
1200 X 0-3 



0-5 



= P. 



Performing the arithmetical operation here indicated, that is, 
multiplying together the two quantities above the line, and 
dividing the product by the quantity under the line, the quo- 
tient will be equal to the quantity represented by P, viz., the 
pressure resisted by the timber, a h. Thus — 

1200 
0-3 



0-5)360-0 

720 = P. 
The strain upon the timber, a 5, is, therefore, equal to 720 
pounds ; and the strain upon the other timber, 5 c , is also 720 
pounds; for in this case, the two timbers being inclined 
equally from the vertical, the line ^ <^ is therefore equal to the 
line 1) e. 




Fig. 207. 



158 AMERICAN HOUSE-CARPENTER. 

261. — In Fig. 207, tlie two supports are inclined at different 
angles, and the pressures are proportionately unequal. The 
supports are also unequal in length. The length of the sup- 
ports does not alter the amount of pressure from the concen- 
trated load supported ; but generally long timbers are not so 
capable of resistance as shorter ones. They yield more readily 
laterally, as they are not so stiff, and shorten more, as the com- 
pression is in proportion to the length. To ascertain the pres- 
sures in Fig. 207, let the weight suspended from 5 ^ be equal 
to two and three-quarter tons, (2*75 tons.) The line & d mea- 
sures five and a half tenths of an inch, (0*55 inch,) and the line 
h 6 half an inch, (0*5 inch.) Therefore, the proportion 
Id : W-heiP, becomes 0*55 : 2-75 :: 0-5 : P, 
. 2-75 X 0-5 o 

2-75 
0-5 



0-55)l-375(2-5 
110 



275 

275 



The strain upon the timber, h e^ is, therefore, equal to two 
and a half tons. 

Again, the line e d measures four-tenths of an inch, (0*4 
inch ;) therefore, the proportion 

hd I Wi\ed: Q, becomes 0*55 : 2-75 :: 04 : Q, 
, 2-75 X 04 ^ 
""^ -0^55- = «• 

2-75 
04 

0-55)l-100(2 = Q, 
110 



FRAMING. 15S 

The strain upon the timber, hf^ is, therefore, equal to two 
ions. 

262. — Thus it is seen that the united pressures exerted by a 
weight upon two inclined supports always exceed the weight. 
In the last case 21 tons exerts a pressure of 2 5 and two tons, 
equal together to 4J tons ; and in the former case, 1200 
pounds exerts a pressure of twice 720 pounds, equal to 1440 
pounds. The smaller the angle of inclination to the horizon- 
tal, the greater will be the pressure upon the supports. So, in 
the frame of a roof, the strain upon the rafters decreases gra- 
dually with the increase of the angle of inclination to the 
horizon, the length of the rafter remaining the same. 

263. — This is true in comparing systems of framing with 
each other; but in a system where the concentrated weight 
to be supported is not in the middle, (see Fig. 207,) and, in 
consequence, the supports are not inclined equally, the strain 
will be greatest upon the support that has the greatest inclina- 
.■»nr> to the horizon. 

^64. — In ordinary cases, in roofs for example, the load is 
not concentrated but is that of the framing itself. Here the 
amount of the load will be in proportion to the length of the 
rafter, and the rafter increases in length with the increase of 
the angle of inclination, the span remaining the same. So it 
is seen that in enlarging the angle of inclination to the horizon 
in order to lessen the oblique thrust, the load is increased in 
consequence of the elongation of the rafter, thus increasing the 
oblique thrust. Hence there is a limit to the angle of inclina- 
tion. A rafter will have the least oblique thrust when its 
angle of inclination to the horizon is 35° 16^ nearly. This 
angle is attained very nearly when the rafter rises 8^ inches 
per foot ; or, when the height, B C^ (Fig, 216,) is to the baae, 
A G, as 8^ is to 12, or as 0*7071 is to 1-0. 

265. — Correct ideas of the comparative pressures exerted 
:apon timbers, according to their position, will be readily 



160 AMERICAJS^ HOUSE-CARPENTEE. 

formed by drawing various designs of framing, and estimating 
the several strains in accordance with the parallelogram of 
forces, always drawing the triangle, h d e, so that the three 
lines shall be parallel with the three forces, or pressures, re- 
spectively. The length of the lines forming this triangle is 
unimportant, but it will be found more convenient if the line 
drawn parallel with the known force is made to contain as 
many inches as the known force contains pounds, or as many 
tenths of an inch as pounds, or as many inches as tons, or 
tenths of an inch as tons : or, in general, as many divisions of 
any convenient scfde as there are units of weight or pressure 
in the known force. If drawn in this manner, then the num- 
ber of divisions of the same scale found in the other two lines 
of the triangle will equal the units of pressure or weight of the 
other two /orces respectively, and the pressures sought will be 
ascertained simply by applying the scale to the lines of the 
triangle. 

For example, in Fig. 207, the vertical line, & 6?, of the tri- 
angle, measures fifty-five hundredths of an inch, (0*55 inch ;) 
the line, 5 ^, fifty-hundred ths, (0*50 inch ;) and the line, e d^ 
forty, (0*40 inch.) l^ow, if it be supposed that the vertical pres- 
sure, or the weight suspended below h d, is equal to 55 pounds, 
then the pressure onh e will equal 50 pounds, and that on e d 
will equal 40 pounds ; for, by the proportion above stated, 
hd : WiiheiF, 
55:55 :: 50:50; 
and so of the other pressure. 

266. — ^If a scale cannot be had of equal proportions with the 
forces, the arithmetical process will be shortened somewhat by 
making the line of the triangle that represents the known 
weight equal to unity of a decimally divided scale, then the 
other lines will be measured in tenths or hundredths ; and in 
the numerical statement of the proportions between the lines 
and forces, the first tenn being unity, the fourth term will bo 



FRAMING. 



161 



ascertained simply by multiplying tlie second and third terms 
together. 

For example, if the three lines are 1, 0*7 and 1*3, and the 
known weight is 6 tons, then 

1) d : W :: d e : P, becomes 
1 : 6 :: 0-7 : P = 4-2, 
equals four and two-tenths tons. Again — 

h d : W :: e d : Q, becomes 
1:6:: 1-3 : Q = 7-8, 
equals seven and eight-tenths tons. 




Fig. 208. 



267. — In Mg. 208 the weight, TT, exerts a pressure on the 
struts in the direction of their length ; their feet, n n, have, 
therefore, a tendency to move in the direction n o, and would 
so move, were they not opposed by a sufficient resistance from 
the blocks, A and A. If a piece of each block be cut off at 
the horizontal line, a n, the feet of the struts would slide away 
from each other along that line, in the direction, n a; but if, 
instead of these, two pieces were cut off at the vertical line, 
n 5, then the struts would descend vertically. To estimate the 
horizontal and the vertical pressures exerted by the struts, let 
n ohe made equal (upon any scale of equal parts) to the num- 

21 



162 



AMERICAN HOUSE-CAIIPENTER. 



ber of tons with which the strut is pressed ; construct the 
parallelogram of forces bj drawing o e parallel to a n^ and of 
parallel toh n / then nf, (by the same scale,) shows the num- 
ber of tons pressure that is exerted by the strut in the direc- 
tion n a, and n e shows the amount exerted in the direction 
n h. By constructing designs similar to this, giving various 
and dissimilar positions to the struts, and then estimating the 
pressures, it will be found in every case that the horizontal 
pressure of one strut is exactly equal to that of the other, how- 
ever much one strut may be inclined more than the other ; 
and also, that the united vertical pressure of the two struts is 
exactly equal to the weight, W. (In this calculation the 
weight of the timbers has not been taken into consideration, 
simply to avoid complication to the learner. In practice it is 
requisite to include the weight of the framing with the load 
upon the framing.) 




Fig. 209. 



268.— Suppose that the two struts, JB and B, {Fig. 208.) 
were rafters of a roof, and that instead of the blocks, A and A^ 
the walls of a building were the supports: then, to prevent 
the walls from being thrown over by the thrust of B and B, 
it would be desirable to remove the horizontal pressure. This 



FRAMING. 



163 



may be done by uniting the feet of the rafters with a rope, 
iron rod, or piece of timber, as in Fig. 209. This figure is 
similar to the truss of a roof. The horizontal strains on the 
tie-beam, tending to pull it asunder in the direction of its 
length, may be measured at the foot of the rafter, as was 
shown at Fig. 208 ; but it can be more readily and as accu- 
rately measured, by drawing from f and e horizontal lines to 
the vertical line, h d^ meeting it in o and o / then f o will be 
the horizontal thrust at B^ and e o at A ; these will be found 
to equal one another. When the rafters of a roof are thus 
connected, all tendency to thrust the walls horizontally is 
removed, the only pressure on them is in a vertical direction, 
being equal to the weight of the roof and whatever it has to 
support. This pressure is beneficial rather than otherwise, as 
a roof having trusses thus formed, and the trusses well braced 
to each other, tends to steady the walls. 




Fig. 211. 



. — Fig, 210 and 211 exhibit methods of framing for sup- 
porting the equal weights, W and W, Suppose it be required 



164 AMERICAN HOTJSE-OARPENTEK. 

to measure and compare the strains produced on the pieces, 
A B and A C. Construct the parallelogram of forces, e hfd, 
according to Art. 258. Then If will show the strain on A B, 
and h e the strain on A C. Bj comparing the figures, 5 d be- 
ing eqnal in each, it will be seen that the strains in Fig. 210 
are about three times as great as those in Fig. 211 : the posi- 
tion of the pieces, A B and A C^ in Fig. 211, is therefore far 
preferable. 




C Fig. 212. 

270. — The Composition of Forces consists in ascertaining the 
direction and amount of one force, which shall be just capable 
of balancing two or more given forces, acting in different 
directions. This is only the reverse of the resolution of forces, 
and the two are founded on one and the same principle, and 
may be solved in the same manner. For example, let A and 
B^ {Fig, 212,) be two pieces of timber, pressed in the direction 
of their length towards h — A by a force equal to 6 tons weight, 
and B equal to 9. To find the direction and amount of pres- 
sure they would unitedly exert, draw the lines^ h e and hf in 
a line with the axes of the timbers, and make h e equal to the 
pressure exerted by B^ viz., 9 ; also make h f equal to the 
pressure on A^ viz., 6, and complete the parallelogram of 
forces, e l f d ; then h d^ the diagonal of the parallelogram, 
will be the direction^ and its length, 9*255 will be the amounty 



FRAMING. 



165 



of the united pressures of A and of B. The line, h d, is 
termed the resultant of the two forces, 5/ and he. If J. and 
JS are to be supported by one post, C, the best position foi 
that post will be in the direction of the diagonal, h d ; and it 
will require to be sufficiently strong to support the united 
pressures of ^ and of j5, which are equal to 9*25 or 9^ tons. 




Fig. 213. 



271. — Another example : let J^ig. 213 represent a piece of 
framing commonly called a crane, which is used for hoisting 
heavy weights by means of the rope. B hf, which passes over 
a pulley at h. This is similar to I^ig. 210 and 211, yet it is 
materially different. In those figures, the strain is in one 
direction only, viz., from h to d / but in this there are two 
strains, from A to B and from A to W. The strain in the 
direction A B is evidently equal to that in the direction A W. 
To ascertain the best position for the strut, A O, make h e 
equal to hf, and complete the parallelogram of forces, e hfd; 
then draw the diagonal, h d, and it will be the position re- 
quired. Should the foot, (7, of the strut be placed either 
higher or lower, the strain on A would be increased. In 
constructing cranes, it is advisable, in order that the piece, 
B A, may be under a gentle pressure, to place the foot of the 



166 



AMERICAN HOUSE-OAEPENTEE. 



strut a trifle lower than where the diagonal, 1) d^ would indi- 
cate, but never higher. 



r\ 



Fig. 214. 




^vvA^ 



272. — Ties and Struts. Timbeiis in a state of tension are 
called ties, while such as are in a state of compression are 
termed struts. This subject can be illustrated in the following 
manner : 

Let A and B, {Fig. 214,) represent beams of timber support- 
ing the weights, TF", W and W ^ A having but one support, 
which is in the middle of its length, and B two, one at each 
end. To show the nature of the strains, let each beam be 
sawed in the middle from a to 5. The effects are obvious : 
the cut in the beam. A, will open, whereas that in B will 
close. If the weights are heavy enough, the beam, A, will 
break at h ; while the cut in B will be closed perfectly tight 
at a, and the beam be very little injured by it. But if, on the 
other hand, the cuts be made in the bottom edge of the tim- 
bers, from G to h, B will be seriously injured, while A will 
scarcely be affected. By this it appears evident that, in a 
piece of timber subject to a pressure across the direction of its 
length, the fibres are exposed to contrary strains. If the tim- 
ber is supported at both ends, as at B, those from the top edge 
down to the middle are compressed in the direction of their 
length, while those from the middle to the bottom edge are in 
a state of tension ; but if the beam is supported as at A, the 
contrary effect is produced ; while the fibres at the middle of 
either beam are not at all strained. The strains in a framed 



FKAMING. 167 

truss are of the same nature as tliose in a single beam. The 
truss for a roof, being supported at each end. has its tie-beam 
in a state of tension, while its rafters are compressed in the 
direction of their length. By this, it appears highly important 
that pieces in a state of tension should be distinguished from 
such as are compressed, in order that the former may be pre- 
served continuous. A strut may be constructed of two or 
more pieces ; yet, where there are many joints, it will not 
resist compression so well. 

273. — To distinguish ties from struts. This may be done 
by the following rule. In Fig. 206, the timbers, a h and h <?, 
are the sustaining forces, and the weight, TF, is the straining 
force; and, if the support be removed, the straining force 
would move from the point of support, 5, towards d. Let it be 
required to ascertain whether the sustaining forces are stretched 
or pressed by the straining force. Hule : upon the directioji 
of the straining force, 2> </, as a diagonal, construct a parallelo- 
gram, ehfd.^ whose sides shall be parallel with the direction 
of the sustaining forces, a h and c d ; through the point, h, 
draw a line, parallel to the diagonal, ef; this may then be 
called the dividing line between ties and struts. Because all 
those supports which are on that side of the dividing line, 
which the straining force would occupy if unresisted, are com- 
pressed, while those on the other side of the dividing line are 
stretched. 

In Fig. 206, the supports are both compressed, being on 
that side of the dividing line which the straining force would 
occupy if unresisted. In Mg. 210 and 211, in wdiich A B and 
A G are the sustaining forces, ^ 6^* is compressed, whereas 
^ ^ is in a state of tension ; A C being on that side of the 
line, h i, which the straining force would occupy if unresisted, 
and A B on the opposite side. The place of the latter might 
be supplied by a chain or rope. In Fig. 209, the foot of the 
I'after at A is sustained by two forces, the wall and the tie- 



168 



AMERICA^^^ HOUSE-CAEPENTEE. 



beam, one perpendicular and the other horizontal : the direc- 
tion of the straining force is indicated bj the line, I a. The 
dividing line, h i^ ascertained bv the rule, shows that the wall 
is pressed and the tie-beam stretched. 




Fig. 215. 

274. — Another example : let EA B F^ {Fig. 215,) represent 
a gate, supported by hinges at A and F. In this case, the 
straining force is the weight of the materials, and the direction 
of course vertical. Ascertain the dividing line at the several 
points, G., B^ /, e/, H and F. It will then appear that the 
•force at G is sustained \)^ A G and G E^ and the dividing 
line shows that the former is stretched and the latter com- 
pressed. The force at J? is supported bj A J5"and H E—ikiQ 
former stretched and the latter compressed. The force at B 
is opposed by H B and A B^ one pressed, the other stretched. 
The force at F is sustained by G F and F E^ G F being 
stretched and i^ jE" pressed. By this it appears that JL ^ is in 
a state of tension, and E F^ of compression ; also, that A H 
and G F are stretched, while B H and G E are compressed : 
which shows the necessity of having A II and G F^ each in 
one whole length, while B H and G E may be, as they are 
shown, each in two pieces. The force at J is sustained by 
G t/and J H, the former stretched and the latter compressed. 



FKAMING. 169 

The piece, G D^ is neither stretched nor pressed, and could be 
dispensed with if the joinings at J and / could be made as 
effectually without it. In case A B should fail, then C D 
would be in a state of tension. 

275. — The centre of gravity. The centre of gravity of a 
uniform prism or cylinder, is in its axis, at the middle of its 
length ; that of a triangle, is in a line drawn from one angle to 
the middle of the opposite side and at one-third of the length 
of the line from that side ; that of a right-angled triangle, at a 
point distant from the perpendicular equal to one-third of the 
base, and distant from the base equal to one-third of the per- 
pendicular ; that of a pyramid or cone, in the axis and at one- 
quarter of the height from the base. 

276. — The centre of gravity of a trapezoid, (a four-sided 
figure having only two of its sides parallel,) is in a line joining 
the centres of the two parallel sides, and at a distance from 
the longest of the parallel sides equal to the product of the 
length into the sum of twice the shorter added to the longer 
of the parallel sides, divided by three times the sum of the 
two parallel sides. Algebraically thus — 

_ l{^ a+h) 
Z{a + l) 
where d equals. the distance from the longest of the parallel 
'Jdes, I the length of the line joining the two parallel sides, 
and a the shorter and h the longer of the parallel sides. 

Example. — A rafter, 25 feet long, has the larger end 14 
inches wide, and the smaller end 10 inches wide, how far from 
the larger end is the centre of gravity located ? 

Here, Z = 25, a = }f , and l = ||, 

_ l&a + h) _ 25 (2 X If -]' H) _ 25 x f | _ 
hence e^_ 3^^^^^ - 3(11 + 11) " "sI^h" ~ 

25 X 34: 850 
o 04 —~^ — ^1'^ = 11 ^^^^ ^8 inches nearly. 

In irregular bodies with plain sides, the centre of gravity 

22 



170 



AMERICAN HOUSE-CARPENTEE. 



may be found by balancing them upon the edge of a prism — 
upon the edge of a table — in two positions, making a line each 
time upon the body in a line with the edge of the prism, and 
the intersection of those lines will indicate the point required. 
Or suspend the article by a cord or thread attaclied to one 
corner or edge ; also, from the same point of suspension, hang 
a plumb-line, and mark its position on the face of the article ; 
again, suspend the article from another corner or side, (nearly 
at right angles to its former position,) and mark the position 
of the plumb-line upon its face ; then the intersection of the 
two lines will be the centre of gr^aTity. 



n 




B 


c II 









Fig. 216. 



277. — The effect of the weight of inclined teams. An in- 
clined post or strut, supporting some heavy pressure applied at 
its upper end, as at Fig. 209, exerts a pressure at its foot in 
the direction of its length, or nearly so. But when such a 
beam is loaded uniformly over its whole length, as the rafter 
of a roof, the pressure at its foot varies considerably from the 
direction of its length. For example, let A B^ (Fig. 216,) be 
a beam leaning against the wall, B c, and supported at its 
foot by the abutment, ^, in the beam, A c, and let <? be the 
centre of gravity of the beam. Through c>, draw the vertical 
line, h d^ and from B^ draw the horizontal line, B J, cutting 
h dinh ; join h and A, and h A will be the direction of the 
thrust. To prevent the beam from loosing its footing, the joint 
at A should be made at right angles to h A. The amount of 
pressure will be found thus: let h d^ (by any scale of equal 



FRAMING. 171 

paits,) equal the niimber of tons up on the beam, A B ; draw 
d e, parallel to B h ; then h e^ (by the same scale,) equals the 
pressure in the direction, h A; and e d^ the pressure against 
the wall at B — and also the horizontal thrust at A^ as these 
are always equal in a construction of this kind. 

278. — The horizontal thrust of an inclined beam, (Fig. 216,) 
— the effect of its own weight — may be calculated thus : 

Bute. — Multiply the weight of the beam in pounds by its 
base, A (7, in feet, and by the distance in feet of its centre of 
gravity, 6», (see Art. 275 and 276,) from the lower end, at A ; 
and divide this product by the product of the length, A B^ 
into the height, B (7, and the quotient will be the horizontal 

thrust in pounds. This may be stated thus : H = -j-j-, where 

d equals the distance of the centre of gravity, o, from the 

lower end ; h equals the base, A C ; w equals the weight of 

the beam ; h equals the height, B C; I equals the length of 

the beam ; and H equals the horizontal thrust. 

Examjple. — A beam, 20 feet long, weighs 300 pounds; its 

centre of gravity is at 9 feet from its lower end; it is so 

inclined that its base is 16 feet and its height 12 feet ; wdiat is 

the horizontal thrust ? 

□- dlw . 9x16x300 9x4x25 „ , . 

Here -^r-i— becomes = = 9x4x5 

hi 12 X 20 5 

= 180 = j5'= the horizontal thrust. 

This rule is for cases where the centre of gravity does not 
occur at the middle of the length of the beam, although it is 
applicable when it does occur at the middle; yet a shorter 
rule will suffice in this case, — and it is thus : — 

Bide. — Multiply the weight of the rafter in pounds by the 
base, A C^ {Fig. 216,) in feet, and divide the product by twice 
the height, B (7, in feet ; and the quotient will be the horizon 
tal thrust, when the cer tre of gravity occurs at the middle of 
the beam. 



172 



AMERICAN HOUSE-CAKPENTER. 



If the inclined beam is loaded with an equally distributed 
load, add this load to the weight of the beam, and use this 
total weight in the rule instead of the weight of the beam. 
And generally, if the centre of gravity of the combined 
weights of the beam and load does not occur at the centre of 
the lenorth of the beam then the former rule is to be used. 




Fig. 217. 



279. — In Fig. 217, two equal beams are supported at their 
feet by the abutments in the tie-beam. This case is similar to 
the last ; for it is obvious that each beam is in precisely the 
position of the beam in Fig. 216. The horizontal pressures at 
B^ being equal and opposite, balance one another ; and their 
horizontal thrusts at the tie-beam are also equal. (See Art. 
2Q'^—Fig. 209.) When the height of a roof, {Fig. 217,) is 
one-fourth of the span, or of a shed, {Fig. 216,) is one-half the 
span, the horizontal thrust of a rafter, whose centre of gravity 
is at the middle of its length, is exactly equal to the weight 
distributed uniformly over its surface. 




'w^'IaJ 



fig 218. 



FKAMING. 1Y3 

280. — In shed, or lean-to roofs, as Fig. 216, tie horizontal 
pressure will be entirely removed, if the bearings of the raft- 
ers, as A B^ {Fig. 218,) are made horizontal — provided, how- 
ever, that the rafters and other framing do not bend between 
the points of support. If a beam or rafter have a natural 
curve, the convex or rounding edge should be laid uppermost. 

281. — A beam laid horizontally, supported at each end and 
uniformly loaded, is subject to the greatest strain at the mid- 
dle of its length. Hence mortices, large knots and other de- 
fects, should be kept as far as possible from that point ; and, 
in resting a load upon a beam, as a partition upon a floor 
beam, the weight should be so adjusted, if possible, that it will 
bear at or near the ends. 

Twice the weight that will break a beam, acting at the 
centre of its length, is required to break it when equally dis- 
tributed over its length ; and precisely the same deflection oi 
sag will be produced on a beam by a load equally distributed, 
that five-eighths of the load will produce if acting at the centre 
of its length. 

282. — When a beam, supported at each end on horizontal 
bearings, (the beam itself being either horizontal or inclined,) 
has its load equally distributed, the amount of pressure caused 
by the load on each point of support is equal to one half the 
load ; and this is also the case, when the load is concentrated 
at the middle of the beam, or has its centre of gravity at the 
middle of the beam ; but, when the load is unequally distri- 
buted or concentrated, so that its centre of gravity occurs at 
some other point than the middle of the beam, then the amount 
of pressure caused by the load on one of the points of support 
is unequal to that on the other. The precise amount on each 
may be ascertained by the following rule. 

Bule. — Multiply the weight w^ {Fig. 219,) by its distance, CB, 
from its nearest point of support, B^ and divide the product 
by the length, A B, of the beam, and the quotient wil be the 



174 



AMEEICAN HOUSE-CAEPENTEE. 





Fig. 219. 



amount of pressure on the remote point of support, A. Again, 
deduct this amount from the weight, ^^, and the remainder 
will be the amount of pressure on the near point of support, 
B ; or, multiply the weight, w^ by its distance, A G^ from the 
remote point of support, A^ and divide the product by the 
length, A B^ and the quotient will be the amount of pressure 
on the near point of support, B. 

When I equals the length, A B ; a = A C ; h = B, and 
to = the load, then 

— T— = A = the amount of pressure at A^ and 



w a 

T 



B = the amount of pressure at B. 



Examjple. — A beam, 20 feet long between the bearings, has 
a load of 100 pounds concentrated at 3 feet from one of the 
bearings, what is the portion of this weight sustained by each 



bearing ? 



Here ^o = 100 ; «, 17 ; J, 3 ; and Z, 20. 
wl 100 X 3 



Hence A 



AndB = 



w a 



20 
100 X 17 



= 15. 



= 85. 



20 

Load on A = 15 pounds. 
Load on B = 85 pounds. 
Total weight = 1( pounds. 



FRAMING. 175 

RESISTANCE OF MATERIALS. 

283. — Before a roof truss, or other piece c f framing, can be 
properly designed, two things are required to be known. The 
one is, the effect of gravity acting npon the various parts of 
the intended structure ; the other, the power of resistance 
possessed by the materials of which the framing is to be con- 
structed. In the preceding pages, the former subject having 
been treated of, it remains now to call attention to the latter. 

284. — Materials used in construction are constituted in their 
structure either of fibres (threads) or of grains, and are termed, 
the former fibrous, the latter granular. Ml woods and wrought 
metals are fibrous, while cast iron, stone, glass, &c., are gra- 
nular. The strength of a granular material lies in the power 
of attraction, acting among the grains of matter of which the 
material is composed, by which it resists any attempt to sepa- 
rate its grains or particles of matter. A fibre of wood or of 
wrought metal has a strength by which it resists being com- 
pressed or shortened, and finally crushed ; also a strength by 
which it resists being extended or made longer, and finally 
sundered. There is another kind of strength in a fibrous mate- 
rial ; it is the adhesion of one fibre to another along their sides, 
or the lateral adhesion of the fibres. 

285. — ^In the strain applied to a piece of timber, as a post 
supporting a weight imposed upon it, (Fig. 220,) we have an 
instance of an attempt to shorten the fibres of which the tim- 
ber is composed. The strength of the timber in this case is 
termed the resistance to comjpression. In the strain on a piece 
of timber like a king-post or suspending piece, [A^ Fig. 221,) 
w'e have an instance of an attempt to extend or lengthen the 
fibres of the material. The strength here exhibited is termed 
the resistance to tension. When a piece of timber is strained 
like a floor beam, or any horizontal piece carrying a load, 
(Fi^. 222,) we have an instance in which the two strains of 



176 



AMEEICAN HOUSE-CAKPENTEE. 






Fig. 222. 

compression and tension are brought into action ; the fibres of 
the npper portion of the beam being compressed, and those of 
the under part being stretched. This kind of strength of tim 
ber is termed resistance to cross strains. In each of these three 
kinds of strain to which timber is subjected, the power of 
resistance is in a measure due to the lateral adhesion of the 
fibres, not so much perhaps in the simple tensile strain, yet to 
a considerable degree in the compressive and cross strains. 
But the power of timber, by which it resists a pressure acting 
compressively in the direction of the length of the fibres, tend- 
ing to separate the timber by splitting off a part, as in the 
case of the end of a tie beam, against which the foot of the 
rafter presses — is wholly due to the Jateral adhesion of the 
fibres. 

286. — ^The strength of materials is that power by which they 
resist fracture,^ while the stiffness of materials is that quality 
which enables them to resist deflection or sagging. A know- 
ledge of their strength is useful, in order to determine their 



FRAMING. 177 

jiinits of size to sustain given weiglits safely ; but a knowledge 
of their stiffness is more important, as in almost all construc- 
tions it is desirable not only that the load be safely sustained, 
but that no appearance of weakness be manifested by any sen- 
sible deflection or sagging. 

I. KESISTANOE TO COMPRESSION. 

287. — ^The resistance of materials to the force of compression 
may be considered in four several ways, viz. : 

1st. When tlie pressure is applied to the fibres longitudi 
nally, and on short pieces. 

2d. When the pressure is applied to the fibres longitudi- 
nally, and on long pieces. 

3d. When the pressure is applied to the fibres longitudi- 
nally, and so as to split off the part pressed against, causing 
the fibres to separate by sliding. 

4:th. When the pressure is applied to the fibres trans- 
versely. 

Posts having their height less than ten times their least side 
will crush before bending ; these belong to the first case : 
while posts, whose height is ten times their least side, or more 
than ten times, will bend before crushing ; these belong to the 
second case. 

288. — In the above first and fourth cases of compression, 
experiment has shown that the resistance is in proportion to 
the number of fibres pressed, that is, in proportion to the area. 
For example, if 5,000 pounds is required to crush a prism with 
a base 1 inch square, it will require 20,000 pounds to crush a 
prism having a base of 2 by 2 inches, equal to 4 inches area ; 
because 4 times 5,000 equals 20,000. Experiment has also 
shown that, in the third case, the resistance is in proportion to 
the area of the surface separated without regard to the form 
of the surface. 

289. — In the second case of compression, the resistance is in 

23 



178 AMERICAN HOTJSE-CARPENTER 

proportion to tlie area of the cross section of the piece, multi 
plied by the square of its thickness, and inversely in propor- 
tion to the square of the length, multiplied by the weight. 
When the piece is square, it will bend and break in the direc- 
tion of its diagonal ; here, the resistance is in proportion to the 
square of the diagonal multiplied by the square of the dia- 
gonal, and inversely proportional to the square of the length 
multiplied by the weight. If the piece is round or cylindrical, 
its resistance will be in accordance with the square of the dia- 
meter multi23lied by the square of the diameter, and inversely 
proportional to the square of the length, multiplied by the 
weight. 

290. — These relations between the dimensions of the piece 
strained and its resistance, have resulted from the discussion 
of the subject by various authors, and rules based upon these 
relations are in general use, yet their accuracy is not fully 
established. Some experiments, especially those by Prof. 
Ilodgkinson, have shown that the resistance is in proportion to 
a less power of the diameter, and inversely to a less power of 
the height ; yet the variance is not great, and inasmuch as the 
material is restricted in the rules to a strain decidedly within 
its limits of resistance, no serious error can be made in the 
use of rules based on the aforesaid relations. 

291. — Experiments, In the investigation of the laws appli- 
cable to the resistance of materials, only such of the relations 
of the parts have been considered as apply alike to wood and 
metal, stone and glass, or other material, leaving to experi- 
ment the task of ascertaining the compactness and cohesion of 
particles, and the tenacity and adhesion of fibres ; those quali- 
ties upon which depend the superiority of one kind of material 
over another, and which is represented in the rules by a constant 
number, each specific kind of material having its own special 
constant^ obtained by experimenting on specimens of that 
peculiar material. 



FRAMING. 



179 



292. — The following table exhibits the results of experiments 
on such woods as are in most common use in this country for 
the pui'pose of construction. The resistance of timber of the 



TABLE I. COMPKESSION. 





^ 


i 


as 


II 

.2 2 


1 




•sf 


Kind of Material. 


I 

CO 


5 ^ 

-a .3 

U 




K P «5 


a 

Is 

1^ 


« ® <u 








Pounds 




Pounds 




Pounds 




White wood. 


•397 


per in. 
2432 


600 


per in. 




per in. 
600 


300 


Mahogany (Baywood), 
Ash, 


•489 

•517 


3527 
4175 


880 
1040 






1300 
2300 


650 
1150 


Spruce, 


. 


•369 


4199 


1050 


470 


160 


500 


250 


Chestnut, . 




•491 


4791 


1200 


690 


230 


950 


475 


White pine, 


. 


•388 


4806 


1200 


490 


160 


600 


300 


Ohio pine, 


. 


•586 


4809 


1200 


388 


130 


1250 


625 


Oak, . 


, 


•612 


5316 


1330 


780 


260 


1900 


950 


Hemlock, . 


. 


•423 


5400 


1350 


540 


180 


600 


300 


Black walnut, 


. 


•421 


5594 


1400 






1600 


800 


Maple, 


. 


•574 


6061 


1515 






2050 


1025 


Cherry, 


. 


•494 


6477 


1620 






1900 


950 


White oak, 


, 


•774 


6660 


1665 






2000 


1000 


Georgia pine. 


. 


•613 


6767 


1700 


510 


170 


1700 


850 


Locust, 


, 


•762 


7652 


1910 


1180 


400 


2100 


1050 


Live oak, . 


. 


•916 


7936 


1980 






5100 


2550 


Mahogany (St. ] 
Lignum vitse, 


)omingo), . 


•837 
1-282 


8280 
8650 


2070 
2160 






4300 
5800 


2150 
2900 


Hickory, . 




•877 


9817 


2450 






3100 


1550 



same name varies much ; depending as it obviously must on 
the soil in which it grew, on its age before and after cutting, 
on the time of year when cut, and on the manner in which it 
has been kept since it was cut. And of wood from the same 
tree, much depends upon its location, whether at the butt or 
towards the limbs, and whether at the heart or at the sap, or 
at a point midway from the centre to the circumference of the 
tree. The pieces submitted to experiment were of ordinary 
good quality, such as would be deemed proper to be used in 
framing. The prisms crushed were 2 inches long, and from 1 
inch to l-J inches square ; some were wider one way than the 



180 AMERICAN HOCSE-CARPENTER. 

other, but all containing in area of cross section from 1 to 2 
inches. There were generally three specimens of each kind. 
The weight given in the table is the average crushing weight 
per superficial inch. 

In the preceding table the first column contains the specific 
gravity of the several kinds of wood, showing their compara- 
tive density. The weight in pounds of a cubic foot of any 
kind of wood or other material, is equal to its specific gravity 
multiplied by 62*5 ; this number being the weight in pounds 
of a cubic foot of water. The second column contains the 
weight in pounds required to crush a ]3rism having a base of 
one inch square ; the pressure applied to the fibres longitudi- 
nally. The third column contains the value of G in the rules ; 
G being equal to one-fourth of the crushing weight in the 
preceding column. The fourth column contains the weight 
in pounds, which, applied to the fibres longitudinally, is 
required to force off a part of the piece, causing the fibres to 
separate by sliding, the surface separated being one inch 
square. The fifth column contains the value of H in the 
rules, ^ being equal to one third of the weight in the preced- 
ing column. The sixth column contains the weight in pounds 
required to crush the piece when the pressure is applied to the 
fibres transversely, the piece being one inch thick, and the 
surface crushed being one inch square, and depressed one 
twentieth of an inch deep. The seventh column contains the 
value of Pin the rules ; P being the weight in pounds applied 
to the fibres transversely, which is required to make a sensible 
impression one inch square on the side of the piece, this being 
the greatest weight that would be proper for a post to be 
loaded with per inch surface of bearing, resting on the side of 
the kind of wood set opposite in the table. A greater weight 
would, in proportion to the excess, crush the side of the wood 
under the post, and proportion ably derange the framing, if not 
cause a total failure. It w^ill be observed that the measure of 



FKAMING. 181 

tnis resistance is nsefiil in limiting tlie load on a post accord- 
ing to the kind of material contained, not in the jpost^ but in 
the timber upon which the post presses. 

293. — In Table 11. are the results of experiments made to 
test the resistance of materials to flexure: first, the flexure 
produced by compression, the force acting on the ends of the 
fibres longitudinally; secondly, the flexure arising from the 
efi'ects of a cross strain, the force acting on the side of the 
fibres transversely, the beams l*eing laid on chairs or rests. 
Of white oak, No. 1, there were eight specimens, of 2 by 4 
inches, and 3|- feet long, seasoned more than a year after they 
were prepared for experiment. Of the other kinds of wood 
there were from three to five specimens of each, of li by 2i 
inches, and from 1| to 2f feet long. Of the cast iron there 
were six specimens, of 1 inch square and 1 foot long; and 
of the wrought iron there were five specimens of American, 
three of f by 2 inclies, and two of l-J inches square, and three 
specimens of common English, ^ by 2 inches ; the eight speci- 
mens being each 19 inches long, clear bearing. In each 
case the result is the average of the stiffness of the several 
specimens. The numbers contained in the second column are 
the weights producing the first degree of flexure in a post or 
strut, where the post or strut is one foot long and one inch 
square; so, likewise, the numbers in the fifth column, and 
which are represented in the rules by E^ are the weights 
required to deflect a beam one inch, where the beam is one 
foot long, clear bearing, and one inch square. — (See remarks 
upon this. Art. (321.) The numbers in the third column are 
equal to one-half of those in the second. The numbers con- 
tained in the fourth column, and represented by n in the 
rules, show the greatest rate of deflection that the material 
may be subjected to without injury. This rate multiplied by 
the length in feet, equals the total deflection within the limits 
of elasticity. 



382 



AMEEICAN HOrSE-CAEPENTER. 





TABLE II.- 


-FLEXURE 












Under 


Under 




Speciflc 


Compression. 


Cross Strain. 


Kind of Material. 












Gravity. 


Pounds pro- 


Yalue of 


Yalue of 


Value of 






ducing the 


j5in 


nin 


E\x 






first degree 
of flexure. 


the Kulcs. 


the Pwules. 


the Rules 


Hemlock, .... 


0-402 


2640 


1320 


008794 


1240 


Spruce, . 




•432 


4190 


2095 


0-09197 


1550 


White pine, 




•407 


2350 


1175 


0-1022 


1750 


Ohio yellow pine, . 




•586 


6000 


3000 


0-049 


1970 


Chestnut, .... 




•52 


7720 


3860 


0-07541 


2330 


White oak, No. 1, . 




•82 






009152 


2520 


White oak. No. 2, . 




■805 


6950 


3475 


0-0567 


2590 


Georgia pine, . 




•755 


9660 


4830 


0^077 23 


2970 


Locust, . 




•863 


10920 


5460 


0-06615 


3280 


Cast iron. 




7-042 






00148 


S0500 


Wrought iron, common English, 


7-576 






0-03717 


45500 


Wrought iron, American, . 


7-576 






0-04038 


51400 



PRACTICAIi RULES FOR C0:MPRESSI0N. 



First Case. 



294. — To find the weight that can be safely sustained by a 
post, when the height of the post is less than ten times the 
diameter if round, or ten times the thickness if rectangular, 
and the direction of the pressure coinciding with the axis. 

Bule I. — Multiply the area of the cross-section of the post, 
in inches, by the value of C in Table I., the product will be 
the required weight in pounds. 

AC=w. (1.) 

Example. — A Georgia pine post is 6 feet high, and in cross- 
section, 8 X 12 inches, what weight will it safely sustain ? 
The area = 8 x 12 = 96 inches ; this multiplied by 1700, the 
value of 6', in the table, set opposite Georgia pine, the result, 
163,200, is the weight in pounds required. It will be observed 
that the weight would be the same for a Georgia pine post of 
any height less than 10 times 8 inches =: 80 inches = 6 feet 8 



FRAMING. 183 

inches, provided its breadth and thickness remain the same, 
12 and 8 inches. 

295. — To find the area of the cross-section of a post to sus 
tain a given weight safely, the height of the post being less 
than ten times the diameter if round, or ten times the least 
side if rectangular ; the pressure coinciding with the axis. 

Rul6 II. — Divide the given weight in pounds by the value of 
(7, in Table I., and the product will be the required area in inches 

5 = A (2.) 

Example. — A weight of 38,400 pounds is to be sustained by 
a white pine post 4 feet high, what must be its area of section 
in order to sustain the weight safely ? Here, 38,400 divided 
by 1200, the value of (7, in Table L, set opposite white pine, 
gives a quotient of 32 ; this, therefore, is the required area, 
and such a post may be 5 x 6*4 inches. To find the least side, 
so that it shall not be less than one-tenth of the height, divide 
the height, reduced to inchesj by 10, and make the least side 
to exceed this quotient. The area, divided by the least side 
so determined, will give the wide side. If, however, by this 
process, the first side found should prove to be the greatest, 
then the size of the post is to be found by Kule YII., YIII., or 
IX. 

296. — If the post is to be round, by reference to the Table 
of Circles in the Appendix, the diameter will be found in the 
column of diameters, set opposite to the area of the post found 
in the column of areas, or opposite to the next nearest area. 
For example, suppose the required area, as just found by the 
example under Rule II., is 32 ; by reference to the column of 
areas, 33*183 is the nearest to 32, and the diameter set opposite 
is 6-5. The post may, therefore, be 6|- inches diameter. 

Second Case. 
297. — To ascertain the weight that can be sustained safely 



184 AMEKICAN HOUSE-CAKPENTEE. 

by a post whose height is, at least, ten times its least side if 
rectangular, or ten times its diameter if round, the direction 
of the pressure coinciding with the axis. 

Hule III. — When the jpost is round the weight may be 
found by this rule : Multiply the square of the diameter in 
inches by the square of the diameter in inches, and multiply 
the product by 0'589 times the value of B, in Table II., divide 
this product by the square of the height in feet, and the quo- 
tient will be the required weight in pounds. 

0'5S9 BD'I)' 0-589 B D" 
w= ^, = J, (3.) 

Excmijple. — What weight will a Georgia pine post sustain 
safely, whose diameter is 10 inches and height 10 feet ? The 
square of the diameter is 100 ; 100 x 100 = 10,000. And 
10,000 by 0-589 times 4830, the value of B, Table II., set 
opposite Georgia pine, ^ 28,448,700, and this divided by 100, 
the square of the height, equals 284,487, the weight required, 
in pounds. 

Bute lY. — If the joost he rectangular the weight is found 
by this rale : Multiply the area of the cross-section of the post 
by the square of the thickness, both in inches, and by the 
value of Bj Table II. Divide the product by the square of 
the height in feet, and the quotient will be the required 
weight in pounds. 

Af B h t^ B 
^ = — A^=-l^- (^-^ 

Example. — What weight will a white pine post sustain 
safely, whose height is 12 feet, and sides 8 and 12 inches re- 
spectively ? The area = 8 x 12 = 96 inches ; the square of 
the thickness, 8, = 64. The area by the square of the thick- 
ness, 96 X 64, = 6144 ; and this by 1175, the value of B^ for 
white pine, equals 7,219,200. This, divided by 144, the 
square of the he'ght = 50,133^, the required weight in 
pounds. 



FRAMING. 



185 



Eule Y.—If the ])Ost le square, the weight is found by this 
rule : Multiply the value of ^, Table IL, by the square of the 
area of the post in inches, and divide the product by the 
square of the height in feet, and the quotient will be the 
required weight in pounds. 

J.2 B D' B .^^ 

^^-A^- = — ^-- ^^^ 

ExaTryple. — ^What weight will a white oak post sustain 
safely, whose height is 9 feet, and sides each 6 inches ? The 
value of B, set opposite white oak, is 3475 ; this, by (36 x 36 
=) 1296, tlie square of the area, equals 4,503,600. This pro- 
duct, divided by 81, the square of the height, gives for quo- 
tient, 55,600, the required weight in pounds. 

298. — ^To ascertain the size of a post to sustain safely a given 
weight when the height of the post is at least ten times the 
least side or diameter. 

Bute YI. — When the post is to le round or cylindrical, the 
size may be obtained by this rule : Divide the weight in 
pounds by 0*589 times the value of B, Table 11., and extract 
the square root of the product ; multiply the square root by 
the height in feet, and the square root of this product will be 
the diameter of the post in inches. 






•589 5 



(6.) 



Example. — What must be the diameter of a locust post, 10 
feet high, to sustain safely 40,000 pounds ? Here 0*589 times 
5,460, the value of B for locust, Table IL, equals 3215*9. 
The weight, 40,000, divided by 3215*9, equals 12*438. The 
square root of this, 3*5268, multiplied by 10, the height, equals 
35*268, and the square root of this is 5*9386 or 5|f inches, the 
required diameter of the post. 

Bute YII. — If the post is to he rectangular, the size may be 
obtained by tliis rule : Multiply the square of the height in 

24 



lvS6 AMERICAN H0L-5E-CARPENTEK. 

feet by tlie weight in pounds, and divide the product by the 
value of B, Table II. ISTow, if the breadth is known, divide 
the quotient by the breadth in inches, and the cube root of 
this quotient will be the thickness in inches. But if the thick- 
ness is known, and the breadth desired, divide, instead, by the 
cube of the thickness in inches, and the quotient will be the 
breadth in inches. 






w 



Bl 

I = ^ (8.) 

Bt' ^ ^ 

Example, — ^What thickness must a hemlock post have, 
whose breadth is 4 inches and height 12 feet, to sustain safely 
1,000 pounds ? The square of the height equals 144 ; this, by 
1,000, the weight, equals 144,000. This, divided by 1,320, the 
value of B for hemlock. Table 11., equals 109*091. This, 
divided by 4, the breadth, equals 27'273, and the cube root 
of this is 3*01, a trifle over 3 inches, and this is the thickness 
required. 

Another Example. — ^What breadth must a spruce post have, 
whose thickness is 4 inches and height 10 feet, to sustain safely 
10,000 pounds? The square of the height, 100, by 10,000, the 
weight, equals 1,000,000. This, divided by 2095, the value of 
B. Table II., for spruce, equals 477*09 ; and this, divided by 
64, the cube of the thickness, equals 7*45, nearly 1^ inches, 
the breadth required. 

Bide Yin. — If the jpost is to he square, the size may be 
obtained by this rule. Divide the weight in pounds by the 
value of ^, Table II., and multiply the square root of the pro- 
duct by the height in feet, and the square root of this product 
will be the dimension of a side of the post in inches. 

Example. — What dimension must the side of a square post 



FEAMING. 187 

have, whose height is 15 feet, the post being of Georgia 
pine, to sustain safely 50,000 pounds? The weight 50,000, 
divided by 4830, the value of B^ Table II., for Georgia pine, 
equals 10-352. The square root of this, 3-2175, multiplied by 
15, the height, equals 48*362, and the square root of this is 
6-9472, nearly 7 inches, the size of a side of the required 
post. 

299. — A square post is not the stiffest that can be made from 
a given amount of material. The stiffest rectangular post is 
that whose sides are in proportion as 6 is to 10. When this 
proportion is desired it may be obtained by the following rule. 

Bule IX. — Divide six-tenths of the weight in pounds by the 
value of j&. Table II., and extract the square root of the quo- 
tient ; multiply the square root by the height in feet, and then 
the square root of this product will be the thickness in inches. 
The breadth is equal to the thickness divided by 0*6. 

/ _ \Ik /^^ v/Q-6 A^ y^ (10.) 

^=o4 (^^-^ 

Example. — What must be the breadth and thickness of a 
white pine post, 10 feet high, to sustain safely 25,000 pounds. 
Here j% of 25,000, the weight, divided by 1175, the value of 
j5. Table IL, for white pine, equals 12*766. The square root 
of this, 3*5729, multiplied by 10, the lieight, equals 35*729, and 
the square root of this is 5*977, nearly 6 inches, the thickness 
required. Th"s, divided by 0-6, equals 10, equals the breadth 
in inches required. 

300. — The sides of a post may be obtained in any desirable 
proportion by Kule IX., simply by changing the decimal 0*6 
to such decimal as will be in proportion to unity as one side is 
to be to the other. For example, if it be desired to have the 
sides in proportion as 10 is to 9, then 0*9 is the required 
decimal ; if as 10 is to 8, then 0*8 is the decimal ; if as 10 



1S8 AilEKICAN HOUSE-CAKPKN^TEK. 

is to 7, then 0*7 is the decimal to be used in place of 0-6 
in the rule. And generally let h equal the broad side and i 
the narrow side, or let these letters represent respectively the 
numbers that the sides are to be in proportion to ; then, where 

X equals the decimal sought, h I t ::1 \ x = — — the required 

decimal, ov fraction. For a fraction may be used in place of 
the decimal, where it would be more convenient, as is the case 
when the sides are desired to be in proportion as 3 to 2. Here 
3 : 2 :: 1 : a? = §. This fraction should be used in the rule in 
place of the decimal 0*6 — rather than its equivalent decimal ; 
simply because the decimal contains many figures, and there- 
fore would not be convenient. The decimal equivalent to f is 

Third Case. 

301. — ^To ascertain what weight may be sustained safely 
by the resistance oF a given area of surface, when the weight 
tends to split off the part pressed against by causing one sm*- 
face to slide on the other, in case of fracture. 

Rule X. — Multiply the area of the surface by the value of 
H^ in Table L, and the product will be the weight required in 
pounds. 

AH=io. (12.) 

Examjple. — The foot of a rafter is framed into the end of 
its tie-beam, so that the uncut substance of the tie-beam is 15 
inches long from the end of the tie-beam to the joint of the 
rafter ; the tie-beam is of white pine, and is six inches thick ; 
what amount of horizontal thrust will this end of the tie-beam 
sustain, without danger of having the end of the tie-beam split 
off? Here the area of surface that sustains the pressure is 6 
by 15 inches, equal to 90 inches. This, multiplied by 160, the 
value of H, set opposite to white pine. Table I., gives a product 
of 14,400, and this is the required weight in pounds. 



FRAMING. 189 

302. — To ascertain tlie area of surface that is required to 
sustain a given weight safely, when the weight tends to split 
off the part pressed against, by causing one surface to slide on 
the other, in case of fracture. 

Rule XL — Divide tlie given weight in pounds by the value 
of i7, Table I., and the quotient will be the required area in 
inches. 

^=|. (13.) 

Example. — ^The load on a rafter causes a horizontal thrust at 
its foot of 40,000 pounds, tending to split off the end of the tie- 
beam, what must be the length of the tie-beam beyond the 
line, where the foot of the rafter is framed into it, the tie-beam 
being of Georgia pine, and nine inches thick ? The weight, or 
horizontal thrust, 40,000, divided by 170, the value of H^ 
Table L, set opposite Georgia pine, gives a quotient of 235-3. 
This, the area of surface in inches, divided by 9, the breadth 
of the surface strained, (equal to the thickness of the tie-beam,) 
the quotient, 26'1, is the length in inches from the end of 
the tie-beam to the rafter joint, say 26 inches. 

303. — A knowledge of this kind of resistance of materials is 
useful, also, in ascertaining the length of framed tenons, so as 
to prevent the pin, or key, with which they are fastened from 
tearing out ; and, also, in cases where tie-beams, or other 
timber under a tensile strain, are spliced, this rule gives the 
length of the joggle on each end of the splice. 

Fourth Case. 

304. — To ascertain what weight a post may be loaded with, 
so as not to crush the surface against which it presses. 

Rule XII. — Multiply the area of the post in inches by the 
value of P, Table I., and the product is the weight required in 
pounds, 

w^AP. (14.) 



190 AMERICAN HOrSE-CAKPEXTEE. 

Example. — A post, S by 10 inches, stands upon a white \ ine 
girder; the area equals 8 x 10 = 80 inches. This, by 300, tlie 
Tahie of P, Table I., set opposite white pine, the product, 
21:,000, is the required weight in poinds. 

305. — To ascertain what area a post must have in order to 
prevent the post, loaded with a given weight, from crushing 
the surface against which it presses. 

Hule Xni. — Divide the given weight in j)ounds by the value 
of P, Table I., and the quotient will be the area required in 
inches. 

A =11. (15.) 

Ecamjple. — A post standing on a Georgia pine girder is 
loaded with 100,000 pounds, what must be its area? The 
weight, 100,000, divided by 850, the value of P, Table I., set 
opposite Georgia pine, the quotient, 117*65, is the required 
area in inches. The post maybe 10 by llf, or 10 x 12 inches, 
or, if square, each side will be 10'S4: inches, or 12J inches 
diameter, if round. 

n. EE5I5TAXCE TO TEXSIOX. 

306. — The resistance of materials to the force of stretching, 
as exemplified in the case of a rope fi-om which a weight is 
suspended, is termed the resistance to tension. In fibrous 
materials, this force will be different in the same specimen, in 
accordance with the dirtction in which the force acts, whether 
in the direction of the length of the fibres, or at right angles to 
the direction of their length. It has been found that, in hard 
woods, the resistance in the former direction is about 8 to 10 
times what it is in the latter; and in soft woods, straight 
grained, such as wliite pine, the resistance is from 16 to 20 
times. A knowledge of the resistance in the direction of the 
fibres is the most useful in practice. 

307.- -In the following table, the experiments recorded were 



FRAMING. 



i9J 



to test this resistance in such woods, also iron, as are in common 
use. Each specimen was turned cylindrical, and about 2 
inches diameter, and then the middle part for 10 inches in 
length reduced to f ths of an inch diameter, at the middle of 
the reduced part, and gradually increased toward each end, 
where it was about an eighth of an inch larger at its junction 
with the enlarged end. 



TABLE ni. TENSION. 





Specific 
Gravity, 


Weight produc- 


Value of 


Kind of Material. 


ing fracture per 
square inch. 


T 




in the Eules. 






Pounds. 




Hickory, 


o-'zsi 


20,700 


3,450 


Locust, . 






•794 


] 5,900 


2,650 


Maple, . 






•694 


15,400 


2,567 


White pine, 






•458 


14,200 


2,367 


Ash, 






•608 


11,700 


1,950 


Oak, 






•728 


10,000 


1,667 


White oak. 






•774 


17,000 


2,833 


Georgia pine, . 






•650 


17,000 


2,833 


Cast iron, ] J^^" 






7-200 
7-600 


17,000 
30,000 


2,833 
5,000 


American wrought iron, 2 in. diam.. 


7*000 


30,000 


5,000 


Do. do. f and ^ do., 


7-800 


55,000 


9,166 


Do. do. wire, No. 3, . 




102,000 


17,000 


Do. do. do. No. 0, . 




74,500 


12,416 


Do. annealed do. No. 0, . 




53,000 


8,833 



308. — ^The value of Tin the rules, as contained in the last 
column of the above table, is one-sixth of the weight pro- 
ducing fracture per square inch of cross section, as recorded in 
the preceding column. This proportion of the breaking 
weight is deemed the proper one, from the fact that in prac- 
tice, through defects in workmanship, the attachments may be 
so made as to cause the strain to act along one side of the 
piece, instead of through its axis ; and as in this case it has 
been found, that fracture will be produced with ^ of the strain 
that can be sustained through the axis, therefore one half of 
this reduced strain, (equal to } of the strain through the axis), 
is the largest that a due regard to security will permit to be 



192 AJklERICAK HOUSE-CAEPENTEE. 

used. And in some cases it may be deemed advisable to load 
the material with even a still smaller strain. 

309. — To ascertain the weight or pressure that may be safely 
applied to a beam as a tensile strain. 

Bule XIY. — Multiply the area of the cross section of the 
beam in inches by the value of T, Table III., and the product 
will be the required weight in pounds. The cross section here 
intended is that taken at the smallest part of the beam or rod. 
A beam is usually cut with mortices in framing ; the area will 
probably be smallest at the severest cutting : the area used in 
the rule must be only of the uncut fibres. 

AT=.w. (16.) 

Example. — A tie-beam of a roof truss is of white pine, and 
6 X 10 inches ; the cutting for the foot of the rafter reduces 
the uncut area to 40 inches: what amount of horizontal thrust 
from the foot of the rafter will this tie-beam safely sustain? 
Here 40 times 2,367, the value of T, equals 94,680, the required 
weight in pounds. 

310. — To ascertain the sectional area of a beam or rod that 
will sustain a given weight safely, when applied as a tensile 
strain. 

Rule XY. — Divide the given weight in pounds by the value 
of T, Table III., and the quotient will be the area required in 
inches : this will be the smallest area of uncut fibres. If the 
piece is to be cut for mortices, or for any other purpose, then 
na sufficient addition is to be made to the result found by 
the rule. 

5= A (17.) 

Example. — A rafter produces a thrust horizontally of 
80,000 pounds ; the tie-beam is to be of oak : what must the 
area of the cross section of the tie-beam be, in order to sustain 
the rafter safely? The given weight, 80,000, divided by 
1,667, the value of 2\ the quotient, 48, is the area of uncut 



FRAMING. 1 93 

fibres. Tliis should have usually one-half of its amount added 
to it as an allowance for cutting ; therefore 48 + 24 = Y2. 
The tie-beam may be 6 x 12 inches. 

311. — In these rules nothing has been said of an allowance 
for the weight of the beam itself, in cases where the beam is 
placed vertically, and the weight suspended from the end. 
Usually, in timber, this is small in comparison with the load, 
and may be neglected; although in very long timbers, and where 
accuracy is decidedly essential, it may form a part of the rule. 

312. — Taking the eifect of the weight of the beam into 
account, the relation existing between the weights and parts 
of the beam, may be stated algebraically thus : — 
AT^w-^h (18.) 

Where A equals the area of the section of uncut fibres, T 
equals the tabular constant in the rules, which is equal to the 
load that may be safely trusted on a rod of like material with 
the beam and one inch square ; w equals the load, and It 
equals the weight of the beam. Now, the w^eight of the beam 
equals its cubical contents in feet, multiplied by the weight of 
a cubic foot of like material ; and a cubic foot of the material 
equals 62-5 times its specific gravity, while the cubical contents 

of the beam in feet equals Z, where It equals the sectional 

IM 

area in inches, and I equals the length in feet. Hence — 
^ = 63-5/j^?, (19.) 

where/* equals the specific gravity. It will be observed that 
A equals the sectional area of the uncut fibres, while R equals- 
the sectional area of the entire beam ; and, where the excess 
of B, over A may be stated as a proportional part of A^ or 
when A -\- n A = R^ {n being a decimal in proportion ta 
unity, as the excess of R over A is to A^) or 

R — A 

—J — = n. Then, [from (18.) ] — 

25 



194 AMEEICAN HOUSE-CARPENTER. 

AT =w + h. 

144 -^ 

= -z^-f 0-434 (n + l)^/Z; 
and 7^ = J. r- 0-434 {n + 1) Afl, 

w = A{T- 0-434 {n^-l)fl ;) (20.) 

When ^ is found, to find ^, we liave from 

I^ = A+nA, 

B=A{n-hl.) (22.) 

A.S the excess of i? over A decreases, n also decreases, until 
finally, when B = A,n becomes zero. For — 

and when ^ = ^, then 

When n equals zero, it disappears from the rules, and (20) 
becomes 

w = A (r- 0-434/ Z) (23.) 

and (21) becomes 

^ = T-olufl, (^*-> 

and (22) becomes 

I^ = A, (25.) 

313. — ^These rules stated in words at length are as fol- 
40WS : — 

To ascertain the weight that may be suspended safely from 

A vertical beam, when the weight of the beam itself is to be 

taken into account, and when a portion of the fibres are cut in 

framing. 

Jiule XYI. — ^From the sectional area of the beam, deduct 



FRAMING. 195 

the sectional area of uncut fibres, and divide the remainder by 
the sectional area of the uncut fibres, and to the quotient add 
unity; multiply this sum by 0'434: times the specific gravity 
of the beam, and by its length in feet ; substract this product 
from the value of T^ Table III., and the remainder, multiplied 
by the sectional area of the uncut fibres, will be the required 
weight in pounds. 

w:=^A {T- 0434: {n + 1)/ 1) (20.) 

Example. — A. white pine beam, set vertically, 5x9 inches 
and 30 feet long, is so cut by mortices as to have remaining 
only 5x6 inches sectional area of uncut fibres : what weight 
will such a beam sustain safely, as a tensile strain ? The uncut 
fibres, 5 X 6 = 30, deducted from the area of the beam, 5x9 
=: 45, there remains 15. This remainder, divided by 30, the 
area of the uncut fibres, the quotient is 0'5. This added to 
unity, the sum is 1*5. This, by 0*4:34 times 0*458, the specific 
gravity set opposite white pine in Table III., and by 30, the 
length of the beam in feet, the product is 8-95. This product, 
deducted from 2,367, the value of Tset opposite white pine in 
Table III., the remainder is 2,358*05. This remainder multi- 
plied by 30, the sectional area of the uncut fibres,,the product, 
70,741*5, is the required weight in pounds. 

314. — When the beam is uncut for mortices or other pur- 
poses, the former part of the rule is not needed ; the weight 
y/ill then be found by the following rule. 

Bule XYII. — Deduct 0*434 times the specific gravity of 
the beam, multiplied by its length in feet, from the value 
of T, Table III. ; the remainder, multiplied by the sectional 
area of the beam in inches, will be the required weight in 
pounds. 

w = A {T-0'4:S4:fl). (23.) 

Example. — A Georgia pine beam, set vertically, is 25 feet 
long and 7x9 inches in sectional area: what weight will 
it sustain safely, as a tensile strain ? By the rule, 0*434 times 



196 AMEKICAl^ HOrSE-CAEPENTEK. 

0*65, the specific gravity of Georgia pme, as in Table III., mul- 
tiplied by 25, the length in feet, the product is Y*05. This 
product, deducted from 2,833, the value of T, Table III., set 
opposite Georgia pine, and the remainder, 2,825*95, multiplied 
by 63, the sectional area, the product, 178,034-85, is the 
required weight in pounds. 

315. — To ascertain the sectional area of a vertical beam that 
v^ill safely sustain a given tensile strain, where the weight of 
the beam itself is to be considered. 

Rule XV III. — ^Where the beam is cut for mortices or other 
purposes, let the relative proportion of the uncut fibres to those 
that are cut, be as 1 is to n, in being a decimal to be fixed on 
at pleasure.) Then to the value of n add unity, and multiply 
ing the sum by 0*434 times the specific gravity in Table III., 
and by the length in feet. Deduct this product from the 
value of T, Table III., divide the given weight in pounds by 
this remainder, and the quotient will be the area of the uncut 
fibres in inches. Add unity to the value of n, as above, and 
multiply the sum by the area of the uncut fibres ; the product 
will be the required area of the beam in inches. 

^ r- 0*434(^+1)/^, ^ '^ 

E = A{n + \\ (22.) 

Example.— A. vertical beam of white oak. 30 feet long, is 
required to resist efiPectually a tensile strain of 80,000 pounds : 
what must be its sectional area ? The relative proportion of 
the uncut fibres is to be to those that are cut as 1 is to 0*4. 
To 0*4, the value oin, add 1* ; the sum is 1*4. This, by 0*434 
times '774, the specific graA^ity of white oak in Table III., and 
by 30, the length, the product is 14*109. This, deducted from 
2,833, the value of Tfor white oak in Table HI., the remainder 
is 2,818*891. The given weight, 80,000, divided by 2,818*891, 
the remainder, as above, the quotient, 28*38, is the area of the 
uncut fibres. This multiplied by the sura of 0*4 and 1', (or 



FRAMING. 197 

the value of n and unity = 1-4,) the pre duct, 39'732, is the 
required area of the beam in inches. 

316. — When the fibres are uncut, then their sectional area 
equals the area of the beam, and may be found by the follow- 
ing rule. 

Bute XIX, — Deduct 0434 times the specifi.c gravity in 
Table III., multiplied by the length in feet, from the value of 
T^ Table III., and divide the weight in pounds by the remain- 
der. The quotient will be the required area in inches. 

Example. — A vertical beam of locust, 15 feet long, fibres all 
uncut, is required to sustain a tensile strain equal to 25,000 
pounds : what must be its area ? Here 0*434: times '794, the 
specific gravity for locust in Table III., multiplied by 15, the 
length in feet, is 5 '17. This, from 2,650, the value of T for 
locust, Table III., the remainder is 2,644*83. The given 
weight, 25,000, divided by 2,644*83, the remainder, as above, 
the quotient, 9*45, will be the required area in inches. 

in. RESISTiLNrOE TO CEOSS-STEAINS. 

317. — A load placed upon a beam, laid horizontal or in- 
clined, tends to bend it, and if the weight be proportionally 
large, to break it. The power in the material that resists this 
bending or breaking, is termed the resistance to cross- strains^ 
or transverse strains. While in posts or struts the material is 
compressed or shortened, and in ties and suspending-pieces it 
is extended or lengthened ; in beams subjected to cross-strains 
the material is both compressed and extended. (See Art 
254.) When the beam is bent, the fibres on the concave side 
are compressed, while those on the convex side are extended. 
The line where these two portions of the beam meet — that is, 
the portion compressed and the portion extended — the hori- 
zontal line of juncture, is termed the neutral line or plane. It 



198 AMERICAN HOUSE-CAEPENTEE, 

is SO called because at this line the fibres are neither com 
pressed nor extended, and hence are under no strain whatever. 
The location of this line or plane is not far from the middle of 
the depth of the beam, when the strain is not sufficient to 
injure the elasticity of the material ; but it removes towards 
the concave or convex side of the beam as the strain is 
increased, until, at the period of rupture, its distance from the 
top of the beam is in proportion to its distance from the bot- 
tom of the beam as the tensile strength of the material is to its 
compressive strength. 

318. — In order that the strength of a beam be injured as 
little as possible by the cutting required in framing, all mor- 
tices should be located at or near the middle of the depth. 
There is a prevalent idea among some, who are aware that 
the upper fibres of a beam are compressed when subject to 
cross-strains, that it is not injurious to cut these top fibres, 
provided that the cutting be for the insertion of another piece 
of timber — as in the case of gaining the ends of beams into 
the side of a girder. They suppose that the piece filled in 
will as effectually resist the compression as the part removed 
would have done, had it not been taken out. J^ow, besides 
the effect of shrinkage, which of itself is quite sufficient to 
prevent the proper resistance to the strain, there is the mecha- 
nical difficulty of fitting the joints perfectly throughout ; and, 
also, a great loss in the power of resistance, as the material is 
so much less capable of resistance when pressed at right angles 
to the direction of the fibres, than when directly with them, as 
the results of the experiments in the tables show. 

319. — In treating upon the resistance to cross-strains, the 
subject is divided naturally into two parts, viz. stiffness and 
strength : the former being the power to resist deflection or 
bending, and the latter the resistance to rupture. 

320. — Resistance to Deflection. "When a load is placed 
upon a beam supported at each end, the beam bends more or 



FRAMING. 199 

less ; the distance that the beam descends u.ider the operatiop 
of the load, measured at the middle of its length, is termed its 
deflection. In an mvestigation of the laws of deflection it has 
been demonstrated, and experiments have confirmed it, that 
while the elasticity of the material remains uninjured by the 
pressure, or is injured in but a small degree, the amount of 
deflection is directly in proportion to the weight producing it, 
and is as the cube of the length ; and, in pieces of rectangular 
sections, it is inversely proportional to the breadth and the 
cube of the depth : or, inversely proportional to the fourth 
power of the side of a square beam or of the diameter of a 
cylindrical one. Or, when I equals the length between the 
supports, w ihQ weight or pressure, h the breadth, d the depth, 
and p the deflection ; then — 

0^ = ^' (26.) 

equals a constant quantity for beams of all dimensions made 
from a like material. Also, 

V^=^^ (27.) 

where s equals a side of a square beam ; and 
I' w _ 
0-589 D'p-^' (^^0 

where J) equals the diameter of a cylindrical beam. The 
constant here is less than in the case of the square and of the 
rectangular beams. It is as much less as the circular beam is 
less stiff than a square beam whose side is equal to the diame- 
ter of the cylindrical one. The constant, E, is therefore mul- 
tiplied by the decimal 0*589. 

321.— It may be observed that ^in (26) and (27) would be 
equal to w, in case the dimensions of the beam and the 
amount of deflection were each made equal to unity ; and in 
(28) equal to w divided by 0-589. That is, when in (26) the 
length is 1, the breadth 1, and the depth 1, then ^ would be 



200 AMEEICAN HOrSE-CAKPENTER. 

equal to the weight that would depress the beam from its ori 
ginal line equal to 1. Thus — 

^ - 5 6Z> - 1 X r X 1 - '^' 
the dimensions all taken in inches except the length, and this 
taken in feet. This is an extreme state of the case, for in most 
kinds of material this amount of depression would exceed the 
limits of elasticity ; and hence the rule -would here fail to give 
the correct relation among the dimensions and pressure. For 
the law of deflection as above stated, (the deflection being 
equal for equal weights,) is true only while the depressions are 
small in comparison wath the length. I^othing useful is, 
therefore, derived from this position of the question, except to 
give an idea of the nature of the quantity represented by the 
constant, E ; it being in reality a measure of the stifihess of 
the kind of material used in comparing one material with 
another. Whatever may be the dimensions of the beam, E^ 
calculated by (26,) will always be the same quantity for the 
same material; but when various materials are used, ^ will 
vary according to the flexibility or stiifness of each particular 
material. For example, ^will be much greater for iron than 
for wood ; and again, among the vai'ious kinds of wood, it w^ill 
be larger for the stiff woods than for those that are flexible. 

322. — If the amount of deflection that would be proper in 
beams used in framing generally, (such as floor beams, girders 
and rafters,) were agreed upon, the rules would be shortened, 
and the labor of calculation abridged. Tredgold proposed to 
make the deflection in proportion to the length of the beam, 
and the amount at the rate of one-fortieth of an inch (= 0*025 
inch) for every foot of length. He was undoubtedly right in 
the manner and probably so in the rate ; yet, as this is a mat- 
ter of opinion, it were better perhaps to leave the rate of de- 
flection open for the decision of those who use the rules, and 
thon it may be varied to suit the peculiarities of each case 



FRAMmG. 20 1 

that may arise. Any deflection within the limits of the elas- 
ticity of the material, may be given to beams used for some 
purposes, while others require to be restricted to that amount 
of deflection that shall' not be perceptible to a casual observer. 
Let n represent, in the decimal of an inch, the rate of deflec- 
tion per foot of the length of the beam ; then the product of n, 
multiplied by the number of feet contained in the length of 
the beam, will equal the total deflection, —nl. E'ow, if ti Z be 
substituted for p in the formulas, (26,) (27) and (28,) they will 
be rendered more available for general use. For example, let 
this substitution be made in (26,) and there results — 

where I is in feet, and 5, d and n in inches ; and for (27) — 

1 
also for (28)— 



E^l^^j:^, (30.) 

s nl s n^ ^ ^ 



^ - 0-589 B'nl~ 0*589 D' n> ^"^^'^ 
where the notation is as before, with also s and D in inches. 
In these formulas, w represents the weight in pounds concen- 
trated at the middle of the length of the beam. If the weight, 
instead thereof, is equally distributed over the length of the 
beam, then, since f of it concentrated at the middle will de- 
flect a beam to the same depth that the whole does when 
equally distributed, {Art, 281,) therefore — ■ 

^=*^> (33.) 



^^-o-.oftflT/)^^' (3^0 



0-589 D' n' 

where w equals the whole of the equally distributed load. 
Again, if the load is borne by more beams than one, laid 
parallel to each other — as, for example, a series or tier of floor 

26 



202 AMERICAN HOUSE- CAEPEXTEK, 

beams — and tlie load is equally distributed over the supported 
surface or floor ; tlien, if/" represents the number of pounds of 
the load contained on each square foot of the floor,, or the 
pounds' weight per foot superficial, and c represents the dis- 
tance in feet between each two beams, or rather the distance 
from their centres, and I the length of the beam in feet, in the 
clear, between the supports at the ends ; then c I will equal 
the area of surface supported by one of the beams, and/* o I 
will represent the load borne by it, equally distributed over its 
length. jS'ow, if this representation of the load be substituted 
for \o in (32,) (33) and (3i) there results— 

i:J-l^ = K^\ ' (86.) 

J, if civ itoi^ ,... 

^ - 0-589 D' n " 0-589 D' n ^"^^'^ 



Practical Pules and Examples, 

323. — To ascertain the weight, placed upon the middle of a 
beam, that will cause a given deflection. 

Bide XX. — Multiply the area of the cross-section of the 
beam by the square of the depth and by the rate of the deflec- 
tion, all in inches ; multiply the product by the value of E^ 
Table 11., and divide this product by the square of the length 
in feet, and the quotient will be the weight in pounds required. 

Exanvple. — ^What weight can be supported upon the middle 
of a Georgia pine girder, ten feet long, eight inches broad, 
and ten inches deep, the deflection limited to three-tenths of 
an inch, or at the rate of 0'03 of an inch per foot of the length ? 
Here the area equals 8 X 10 = 80 ; the square of the depth 
equals 10 x 10 = 100 : 80 x 100 = 8,000 ; this by 0-03, the 
rate of deflection, the product is 2^0 ; and this by 29T0, the 
value of ^for Georgia pine. Table 11., equals 712,800. This 



FRAMING. W6 

product, divided by 100, the square of the length, the quo! ient, 
7,128, is the weight required in pounds. 

liule XXL — Where the beam is square the weight may be 
found by the preceding rule or by this : — Multiply the square 
of the area of the cross-section by the rate of deflection, both 
in inches, and the product by the value of ^, Table II., and 
divide this product by the square of the length in feet, and the 
quotient will be the weight required in pounds. 

Examjple. — What weight placed on the middle of a spruce 
beam will deflect it seven-tenths of an inch, the beam being 
20 feet long, 6 inches broad, and 6 inches deep ? Here the 
area is 6 x 6 = 36, and its square is 36 X 36 == 1296 ; the rate 
of deflection is equal to the total deflection divided by the 

length, = ^ = 0-035 ; therefore, 1296 x 0*035 = 45-36, and 

this by 1550, the value of E for spruce. Table II., equals 
70,308. This, divided by 400, the square of the length, equals 
175*77, the required weight in pounds. 

Rule XXII. — When the beam is round find the weight by 
this rule : — Multiply the square of the diameter of the cross- 
section by the square of the diameter, and the product by the 
rate of deflection, all in inches, and this product by 0*589 
times the value of E^ Table II. This last product, divided by 
the square of the length in feet, will give the required weight 
in pounds. 

Example. — What weight on the middle of a round white 
pine beam will cause a deflection of 0*028 of an inch per foot, 
the beam being 10 inches diameter and 20 feet long? The 
square of the diameter equals 10 x 10 = 100 ; 100 x 100 = 
10,000 ; this by the rate, 0*028, = 280, and this by 0*589 x 
1750, the value of ^, Table IL, for white pine, equals 288,610. 
This last product, divided by 400, the square of the length, 
equals 721*5, the required weight m pounds. 

324. — To ascertain the weight that will produce a given de 



204 AMEKICAU^ HOUSE-CAEPENTEE. 

flection, when the weight is equally distributed over the length 
of the beam. 

Rule XXIII. — The rules for this are the same as the three 
preceding rules, with this modification, viz., instead of the 
square of the length, divide by five-eighths of the square of 
the length. 

325. — In a series or tier of beams, to ascertain the weight 
per foot, equally distributed over the supported surface, that 
will cause a given deflection in the beam. 

Rule XXIY. — The rules for this are the same as Kules XX., 
XXL, and XXIL, with this modification, viz., instead of the 
square of the length, divide by the product of the distance 
apart in feet between each two beams, (measured from the 
centres of their breadths,) multiplied by five- eighths of the 
cube of the length, and the quotient will be the required 
weight in pounds that may be placed upon each superficial 
foot of the floor or other surface supported by the beams. In 
this and all the other rules, the weight of the material com- 
posing the beams, floor, and other parts of the constructions is 
understood to be a part of the load. Therefore from the ascer- 
tained weight deduct the weight of the framing, floor, plaster- 
ing, or other parts of the construction, and the remainder will 
be the neat load required. 

Exam/pie. — In a tier of white pine beams, 4 x 12 inches, 20 
feet long, placed 16 inches or 1^ feet from centres, what 
weight per foot superficial may be equally distributed over 
the floor covering said beams — the rate of deflection to be not 
more than 0*025 of an inch per foot of the length of the beams. 
Proceeding by Rule XX. as above modified, the area of the 
cross-section, 4 x 12, equals 48 ; this by 144, the square of the 
depth, equals 6912, and this by 0*025, the rate of deflection, 
equals 172*8. Then this product, multiplied by 1750, the 
value of E, Table II., for white pine, equals 302,400. The 
distance between the centres of the beams is Is feet, the cube 



FEAMING. 205 

of tlie length is 8,000, and § by I of 8,000 equals G,666^. The 
above 302,400, divided by 6,6661, the quotient, 45-36, equals 
the required weight in pounds per foot superficial. The 
weight of beams, floor plank, cross-furring, and plastering oc- 
cuiTing under every square foot of the surface of the floor, is 
now to be ascertained. Of the timber in every 16 inches by 
12 inches, there occurs 4 x 12 inches, one foot long ; this 
equals one-third of a cubic foot. I^ow, by proportion, if 16 
inches in width contains -J of a cubic foot, what will 12 inches 

i X 12 12 

in width contain ? ^ ^^ — r. r^ = -^\ = ^ oi 2i cubic foot. 

-Lo o X JLo 

The floor plank (Georgia pine) is 12 x 12 inches, and IJ inches 

li . ^ 

thick, equal to ^77 of a cubic foot, equals :j^, equals 4^8-* C)f the 

furring strips, 1x2 inches, placed 12 inches from centres, 
there will occur one of a foot long in every superficial foot. 
Now, since in a cubic foot there is 144 rods, one inch square 
and one foot long, therefore, this furring strip, 1 x 2 x 12 
inches, equals t!? = tV ^^ ^ cubic foot. The weight of the 
timber and furring strips, being of white pine, may be esti- 
mated together : { + V2 = tI + tV = H ^^ ^ cubic foot. White 
pine varies from 23 to 30 pounds. If it be taken at 30 pounds, 
the beam and furring together will weigh 30 x 4| pounds, 
equals 7*92 pounds. Georgia pine may be taken at 50 pounds 
per cubic foot ;* the weight of the floor plank, then, is 50 X 
/e — 5*21 pounds. A snperflcial foot of lath and plastering 
will weigh about 10 lbs. Tlius, the white pine, T'92, Georgia 
pine, 5*21, and the plastering, 10, together equal 23*13 pounds ; 
this from 45'36, as before ascertained, leaves 22*23, say 22i 
pounds, the neat weight per foot superficial that may be 
equally distributed over the floor as its load. 



* To get the freight of wood or any other material, multiply its specific gravity by C2o, For Spo 
ciflc Graviiies see Tables I., U,, and IH. and the Appendix for Weight of Materials. 



206 



AMERICAN HOUSE-CARPENTER. 



326. — To ascertain the weight when the beam is laid not 
horizontal, but inclined. 

Rule XXY. — In each of the foregoing rules, multiply the 
result there obtained by the length in feet, and divide the pro- 
duct by the horizontal distance between the supports in feet^ 
and the quotient will be the required weight in pounds. 

The foregoing Rules, stated algebraically, are placed in the 
following table : — 



TABLE IV. STIFFNESS OF BEAMS I WEIGHT. 



When the 


When the weight is 


When the beam is 


beam is laid 


Eect- 
angular. 


Square. 


Eoimd. 


norizontal 


Concentrated at middle, w, in pounds, equals 
Equally distributed, ic, in pounds, equals 
By the foot superficial, j^ in pounds, equals 
Concentrated at middle, to, in pounds, equals 

Equally distributed, w, in pounds, equals 
By the foot superficial,/ in pounds, equals 


(3S.) 
Enhd^ 


(89.) 

Ens* 


(40.) 
•589 EnB* 


Horizontal 


(41.) 
Enld-^ 


(42.) 
Ens^ 


(43.) 
•U2AEnDi 


Horizontal 


(44.) 
Enhd^ 

%gU 


(45.) 
Ens"^ 


(46.) 


Inclining 


(47.) 

Enld^ 

Ih 


(48.) 

Ens* 

Ik 


(49.) 
•589 EnD* 
• Ih 


Inclining 


(50.) 

Enhd- 

%lh 


(51.) 
Ens* 


(52.) 

•9424 ^TtZH 

Ih 


Inclining 


(53.) 
En & di 


(54) 
Ens* 


(55.) 

•9424 EnD 

cPh 



In the above table, I equals the breadth, and d equals the 
depth of cross-section of beam ; s equals the breadth of a side 
of a square beam, and D equals the diameter of a round 
beam ; n equals the rate of deflection per foot of the length ; 



FRAMING. 207 

i>, s, 5, d and n^ all m inches ; I equals the length, c equals 
the distance between two parallel beams measured from the 
centres of their breadth ; A equals the horizontal distance 
between the supports of an inclined beam ; I, c and h in feet ; 
w equals the weight in pounds on the beam ; / equals the 
weight upon each superficial foot of a floor or roof supported 
by two or more beams laid parallel and at equal distances 
apart ; ^ is a constant, the value of which is found in Table 
II. ; r is any decimal, chosen at pleasure, in proportion to 
unity, as h is to d, from which proportion h equals d r. 

327. — ^To ascertain the dimensions of the cross-section of a 
beam to support the required weight with a given deflection. 

I^ule XXYI. — Preliminary. When the weight is concen- 
trated at the middle of the length. Multiply the weight in 
pounds by the square of the length in feet, and divide the pro- 
duct by the product of the rate of deflection multiplied by the 
value of jK, Table II., and the quotient equals a quantity which 
may be represented by Ji^— referred to in succeeding rules. 

^ = i/". (56.) 

An ^ ^ 

Rule XXYII. — Preliminary. When the weight is equally 

distributed over the length. Multiply five-eighths of the weight 

in pounds by the square of the length in feet, and divide the 

product by the rate of deflection multiplied by the value of E^ 

Table 11., and the quotient equals a quantity which may be 

represented by I^- — referred to in succeeding rules. 

Bule XXYIII.— Preliminary. When the weight is given 
per foot superficial and supported hy two or more heams. 
Multiply the distance apart between two of the beams, (mea- 
sured from the centres of their breadth,) by the cube of the 
length, both in feet, and multiply the product by ^ve-eighths 
of the weight per foot superficial ; divide this product by the 



208 AMEEICAN HOUSE-CAKPEXTEE. 

product of the rate of deflection, multiplied by the value of E 
Table IL, and tbe quotient equals a quantity which may be 
represented bv TI- — referred to in succeeding rules. 

'if^=Tr. (58.) 

Rule XXIX. — Preliminary. Wlien the 'beam is laid not 
horizontal^ hut inclining. In Rules XXYI. and XXYIL. 
instead of the square of the length multiply by the length, and 
by the horizontal distance between the supports, in feet. And 
in Eule XXYIII., instead of the cvhe of the lengthy multiply 
by the square of the lengthy and by the horizontal distance 
between the supports, in feet. 



From (56) 
From (57) 
From (58) 



'^=M,. (59.) 

ifl^ = .;. (60.) 



i^^=^. (61.; 



Bide XXX. — When the learn is rectangular to find the 
dimensions of the cross-section. Divide the quantity repre- 
sented by JT, ]^ or 17^ (in preceding prehminary rules,) by 
the breadth in inches, and the cube root of the quotient -^vill 
equal the required depth in inches. Or, divide the quantit}^ 
represented by jl/, ^or U^ by the cube of the depth in inches, 
and the quotient will equal the required breadth in inches. 
Or, again, if it be desired to have the breadth and depth in 
proportion, as r is to unity, (where r equals any required deci- 
mal,) divide the quantity represented by JIT, JS' or Z7, by the 
value of r, and extract the square root of the quotient : and 
the square root extracted the second time, will equal the depth 
in inches. MuUiply the depth thus found by the value of r, 
and the product will equal the breadth in inches. 



FKAMING. 209 

Example. — To find the depth. A bepin of spruce, laid on 
supports with a clear bearing of 20 feet, is required to support 
a load of 1674 pounds at the middle, and tlie deflection not to 
exceed 0-05 of an inch per foot ; what must be the depth when 
the breadth is 5 inches. Bj Kule XXYL for load at middle : 
the product of 1674, the weight, by 400, the square of the 
length, equals 669,600. The product of 0-05, the rate of de- 
flection, multiplied by 1550, the value of ^, from Table II., 
set opposite spruce, is 77'5. The aforesaid product, 669,600, 
divided by 77*5, equals 8640, the value of M. Then by Kule 
XXX., 8640, the value of Jf, divided by 5, the breadth, the 
quotient is 1728, and 12, the cube root of this, found in the 
table of the Appendix, equals the required depth in inches. 

Example. — To find the hreadth. Suppose that in the last 
example it were required to have the depth 13 inches ; in that 
case what must be the breadth ? The value of M^ 8640, as 
just found, divided by 2197, the cube of the depth, equals 
3*9326, the required breadth — ^nearly 4 inches. 

Example. — To find hoth Ireadth and dejpth., and in a certain 
proportion. Suppose, in the above example, that neither the 
breadth nor the depth are given, but that they are desired to 
be in proportion as 0*5 is to I'O. ISTow, having ascertained the 
value of Jlf, by Eule XXYL, to be 8640, as above, then, by 
Eule XXX., 8640, divided by 0*5, the ratio, gives for quotient 
17,280. The square root of this (by the table in the Appen- 
dix,) is 131*45, and the square root of this square root is 11465, 
the required depth. The breadth equals 11*465 x 0*5, which 
equals 5*7325. The depth and breadth may be 11^ by 5f 
inches. In cases where the load is equally distributed over 
the length of the beam, the process is precisely the same as set 
forth in the three preceding examples, except that fiwe-eighths 
of the weight is to be used in place of the xohole weight ; and 
hence it would be a useless repetition to give examples to 
illustrate such cases. 

27 



210 AMERICAN HOUSE-CARPENTEK. 

Exam/ple. — When the weight is per foot superficial to find 
tlve depth. A floor is to be constructed to support 600 pounds 
on every superficial foot of its surface. Tlie beams to be of 
white pine, 16 feet long in the clear of the supports or walls, 
placed 16 inches apart, from centres, to be 4 inches thick, and 
the amount of deflection not objectionable provided it be within 
the limits of elasticity. Proceeding by Rule XXYIIL, the 
product of l-J feet, (equal to 16 inches,) multiplied by 4096, the 
cube of the length, equals 5461 -J-. Tliis, multiplied by 312-5, 
(equal to f of the weight,) equals 1,Y06,666. The largest rate 
of deflection within the limits ot the elasticity of white pine is 
0-1022, as per Table II. This, multiplied by 1750, the value of 
^for white pine. Table IL, equals 178*85. The former product, 
1,706,666, divided by the latter, 178-85, equals 9,542-5, the 
value of U, l^ow, by Eule XXX., this value of Z7, 9,542*5, 
divided by 4, the breadth, equals 2385*6, the cube root of 
which, 13-362, is the required depth — nearly 13f inches. 

Example. — To find the hreadth. Suppose, in the last exam- 
ple, that the depth is known but not the breadth, and that the 
depth is to be 13 inches. Having found the value of Z7, as 
before, to be 9542*5, then by Eule XXX., dividing 9542*5, the 
value of U^ by 2197, the cube of the depth, gives a quotient of 
4-3434 and this equals the breadth — nearly 4f inches. 

Example. — To find the depth and Ireadth in a given propor- 
tion. Suppose, in the above example, that the breadth and 
depth are both unknown, and that it is desired to have them 
in proportion as 0*7 is to 1-0. Having found the value of U^ 
as before, to be 9542-5, then by Eule XXX., dividing 9542*5, 
the value of Z7, by 0*7, the quotient is 13,632, the square root 
of which is 116*75, and the square root of this is 10*805, the 
depth in inches. Then 10*805, multiplied by 0*7, the product, 
7*5635, is the breadth in inches. The size may be 7xV ^J l^il 
inches. 

328.— Example. — In the case of inclined heamis to fim^d the 



FKAMING. 2j.1 

'hx^tJi, A beam of white pine, 10 feet long in the clear of the 
bearing, and laid at such an inclination that the horizontal 
distance between the supports is 9 feet, is required to support 
12,000 pounds at the centre of its length, with the greatest 
allowable deflection within the limits of elasticity ; what must 
be its depth when its breadth is fixed at 6 inches ? By refer- 
ence to Table IT. it is seen that the greatest value of n^ within 
the limits of elasticity, is 0-1022. By Kule XXYL, for con- 
centrated load, and Rule XXIX., for inclined beams, 12,000, 
the weight, multiplied by 10, the length, and by 9, the hori- 
zontal distance, equals 1,080,000. The product of 01022, the 
greatest rate of deflection, by 1750, the value of E, Table IT., 
for white pine, equals 178-85. Dividing 1,080,000 by 178*85, 
the quotient is 6038-58, the value of M. Kow, by Rule 
XXX., for rectangular beams, 6038*58, the value of Jf, divid- 
ed by 6. the breadth, the quotient is 1006-43. The cube root 
of this, 10*02, a trifle over 10 inches, is the depth required. 

Example. — In case of inclined heams to find the hreadth. 
In the last example suppose the depth fixed at 12 inches ; 
then by Rule XXX., 6038*58, the value of M^ as above found, 
divided by 1728, the cube of the depth, equals 3-494:5, or 
nearly 3^ inches — the breadth required. 

Examjple. — Again, in case the hreadth and depth are to he in 
a certain proportion '^ as, for example, as 0-4 is to unity. 
Then by Rule XXX., 6038*58, the value of J!/", found as above, 
divided by 0*4, equals 15,096*45, the square root of which is 
122*87, and the square root of this square root is 11-0843, a 
trifle over 11 inches — the depth required. Again, 11 multi- 
plied by the decimal 0*4, (as above,) equals 4-4, a little over 
4f inches — the breadth required. 

In the three preceding examples, the weight is understood 
to be concentrated at the middle. If, however, the weight 
had been equally distributed, the same process would have 
been used to obtain the dimensions of the cross-section, with 



212 AMERICAN HOUSE-CAEPENTEK. 

only one exception : viz. f of the weight instead of the whole 
weight wonld have been used. (See Eule XXYII.) 

Example. — In case of inclined hearas ; the weight per foot 
superficial^ and home hy two or more heams. A tier of spruce 
beams, laid with a clear bearing of 10 feet, and at 20 inches 
apart from centres, and laid so inclining that the horizontal 
distance between bearings is 8 feet, are required to sustain 40 
pounds per superficial foot, with a deflection not to exceed 
02 inch per foot of the length ; what must be the depth 
when the breadth is 3 inches ? Proceeding by Eule XXIX. 
for inclined beams, and by Rule XXYIII., If, (=20 inches,) 
the distance from centres, multiplied by 100, the square of the 
length, and by 8, the horizontal distance between bearings, 
equals 1,3334 ; this, by f x 40, five-eighths of the weight, 
equals 33,333^. This, divided by 0*02 x 1550, the rate of 
deflection, by the value of E, Table II., for spruce, equal to 
31, equals 1075-27, the value of U. ISTow by Rule XXX. for 
rectangular beams, 1075*27, divided by 3, the breadth, equals 
35842, the cube root of which, 7'1, is the required depth in 
inches. 

Example. — The same as the preceding i hut to find the 
hreadths when the depth is jhxed at 8 inches. By Rule XXX., 
1075-27, the value of U, diWded by 512, the cube of the 
depth, equals 2-1 — tlie breadth required in inches. 

Example. — The same as the next hut one preceding ; hut to 
find the hreadth and depth in the proportion of 0'^ to 1*0, or 
3 to 10. By Rule XXX., 1075-27, the value of U, divided by 
0-3, the value of r, equals 3584*23. The square root of this is 
59-869, and the square root of this square root is 7*737 — the 
depth required in inches. This 7 '737, multiplied by 0-3, the 
value of r, equals 2*3211 — the required breadth in inches. 
The dimensions may, therefore, be ^j^^ by 71 inches. 

Rule XXXI. — When the heam is square to find the side. 
Extract the square root of the quantity represented by J/", N 



FKAMmG. 213 

or TI, in preliminary Rules XXYI., XXYII. and XXYIII., 

and the square root of tliis square root will equal the side 
required. 

Example. — A beam of chestnut, having a clear bearing of 8 
feet, is required to sustain at the middle a load of 1500 
pounds ; what must be the size of its sides in order that the 
deflection shall not exceed 0*03 inch per foot of its length ? By 
Rule XXYI., 1500, the load, multiplied by 64, the square of 
the length, equals 96,000. This product divided by 0-03 times 
2330, the value of E^ Table IL, for chestnut, gives a quotient 
of 1373'4:, the quantity represented hj 3L ISTow by Rule 
XXXL. the square root of 13734 is 37*05, and the square 
root of this is 6'087. The beam must, therefore, be 6 inches 
square. In this example, had the load, instead of being con- 
centrated at the middle, been equally distributed over its 
length, the side would have been equal to the side just found, 
multiplied by the fourth root of f or of 0'625, equal to 6'087 
X 0-889 =: 54 inches. (See Rules XXYII. and XXXI.) 

Example. — In the case where the weight is per foot superfi- 
cial and home hy two or more heavis. A floor, the beams of 
which are of oak, and placed 20 inches or 1^ feet apart from 
centres, and which have a clear bearing of 20 feet, is required 
to sustain 200 pounds per superficial foot, the deflection not to 
exceed 0*025 inch per foot of the length, and the beam to be 
square. By Rule XXV 111., If, the distance from centres, 
multiplied by 8000, the cube of the length, equals 18,333 § ; 
and this by 125, (being I of 200 pounds,) equals l^QQ^fiQ^l. 
Dividing this by 0*025 times 2520, the value of E, Table IL, 
for oak, the quotient is 26,455 — a number represented by U. 
Now by Rule XXXL, the square root of this number is 162*65. 
and the square root of this square root is 12*753 — the required 
side. The beam may be 12f inches square. 

Example. — Inclined square learns^ load at middle. A bar 
of cast-iron, 6 feet long in the clear of bearings, and laid 



214 AMERICAN HOUSE-CAEPEN^IEE. 

inclining so that the horizontal distance between the bearings 
is 5 feetj is required to sustain at the middle 3000 pounds, and 
the deflection not to exceed O'Ol inch per foot of its length ; 
what must be the size of its sides ? 

By Kule XXYI. for load at middle, modified by Eule 
XXIX. for inclined beams ; 3000, the weight, multiphed by 6, 
the length, and by 5, the horizontal distance between bear- 
ings, equals 90,000. The rate of deflection, O'Ol, by 30,500, 
the value of E^ Table II., for cast-iron, equals 305 ; and 9000 
divided by 305, equals 295-082, the value of M. Now by Paile 
XXXI. for square beams, the square root of 295-082 is IT'18, 
the square root of which is 4*145 — the size of the side required ; 
a trifle over 48, the bar may, therefore, be 4|- inches square. 

Example. — Same as preceding^ hut the weight equally distri- 
huted. By Rule XXYII. f of the weight is to be used instead 
of the weight ; therefore 295*082, the value of Jf, as above, 
multiplied by I, will equal 184*426, the value of iV^. By Eule 
XXXI. the square root of 184*426 is 13*58, the square root of 
w^hich is 3*685 — the size of the side required ; equal to nearly 
3j^ inches square. 

Example. — Same as preceding case^ hut the weight per foot 
superficial^ and sustained by 2 or more bars, placed 2 feet 
from centres, the load being 250 pounds per foot superficial. 
By Rule XXYIIL, modified by Rule XXIX., the distance 
from centres, 2, multiplied by 36, the square of the length, 
and by 5, the horizontal distance, equals 360. This by 156*25, 
five-eighths of the weight, equals 56,250. The rate of deflec- 
tion, 0*01, by 30,500, the value of E^ Table II., for cast-iron, 
equals 305. The above 56,250, divided by 305, equals 184*426, 
the value of TJ. ^Now by Rule XXXI. the square root of 
184*426. the value of C^, is 13*58, the square root of which is 
3*685 — the size of the side required. It will be observed that 
this result is precisely like that in the last example. This is as 
it should be, for each beam has to sustain the weight on 2 x 6 



FRAMING. 215 

^ 12 superficial feet, equal to 12 x 250, equal 3( 00 pounds ; 
aud all the other conditions are parallel. 

Idule XXXII. — When the heam is round to find the diame- 
ter. Divide the value of J!/", iTor Z7, found by Eules XXYL, 
XXYII. or XXYIII., bj the decimal 0*589, and extract the 
square root : and the square root of this square root will be 
the diameter required. 

Examjple. — In the case of a concentrated load at middle. A 
round bar of American iron, of 5 feet clear bearing, is required 
to sustain 800 pounds at the middle, with a deflection not to 
exceed 0*02 inch per foot ; what must be its diameter ? By 
Eule XXYL for load at middle, 800, the weight, multiplied 
by 25, the square of the length, equals 20,000. The rate of 
deflection, 0*02, by 51,400, the value of ^, Table IL, for Ame- 
rican wrought iron, equals 1028. The above 20,000, divided 
by 1028, equals 19-4:552, the value of M. Now, by Eule 
XXXn., 194552, the value of Jf, divided by 0-589 equals 
33-03, the square root of which is 5-74:7, and the square root 
of this is 2*397, nearly 2-4, the diameter required in inches, 
equal to 2f large. 

Example. — Same case as the preceding^ hut the load equally 
distributed. By Eule XXYII., five-eighths of the weight is to 
be used instead of the w^hole weight ; therefore the above 
33*03, multiplied by I, equals 20-64375, the square root of 
which is 4-544, and the square root of this square root is 2-132, 
the diameter required in inches, 2-J inches large. 

Examjple. — When. the weight is jper foot superficial.^ and sus- 
tained hy two or more hars or heams. The conditions being 
the same as in the preceding examples, but the weight, 100 
pounds per foot, is to be sustained on a series of round rods, 
placed 18 inches apart from centres, equal 1*5 feet. By Eule 
XXYIII., for weight per foot superficial, 1-5, the distance 
from centres, multiplied by 125, the cube of the length, and 
by 62-5, five-eighths of the weight, equals 11,718-75. This 



216 AMERICAN HOIJSE-CAEPENTEE. 

divided by 1028, the product of the rate of deflection by the 
vahie of E^ as found in the preceding example, equals 11*4, 
the value of TI. Now by Eule XXYIL, 114, the vahie of U, 
divided by 0*589, equals 1942, the square root of which is 
4*4:07, and the square root of this square root is 2*099, th.e 
diameter required — very nearly 2yV inches. 

Example. — When the 'beam is round and laid inclining^ the 
weight concentrated at the middle. A round beam of white 
pine, 20 feet long between bearings, and laid inclining so that 
the horizontal distance between bearings is 18 feet, is required 
to support 1250 pounds at the middle, with a deflection not to 
exceed 0*05 inch per foot ; what must be its diameter ? By 
Eule XXYI. for load at middle, modified by Kule XXIX. for 
inclined beams, 1250, the weight, multiplied by 20, the length, 
and by 18, the horizontal distance, equals 450,000. Tlie rate 
of deflection, 0*05, multiplied by 1750, the value of E^ Table 
XL, for white pine, equals 87*5. The above 450,000 divided 
by 87*5, equals 5142*86, the value of M. 1^o\y by Eule 
XXXII. for round beams, 5142*86, the value of if, divided by 
0-589, equals 8731*5, the square root of which is 93*44, and 
the square root of this square root is 9*667, the diameter re- 
quired — equal to 9§ inches. 

Example. — Same as in preceding example^ 'but the weight 
equally dist/ributed. By Eule XXYII., five-eighths of the 
weight is to be used instead of the whole weight, therefore 
8731*5, the result in the last example just previous to taking 
the square root, multiplied by f , equals 5457*2, the square root 
of which is 73*87, and the square root of this square root is 
8*59, the diameter required — nearly 8f inches. 

Example. — Bame as in the next but one preceding example-^ 
but the weight 'per foot superficial., and supported by two or 
more beams. A series of round hemlock poles or beams, 10 
feet long clear bearing, laid inclining so as that the horizontal 
distance between the supports equals 7 feet, and laid 2 feet 



FRAMING. 217 

and 6 inches apart from centres, are required to support 20 
pounds per superficial foot without regard to the amount of 
deflection, provided that the elasticity of the material be not 
injured ; what must be their diameter ? By Rule XXYIII. 
for weight per foot superficial, modified by Rule XXIX. for 
inclined beams, 2*5, the distance from centres, multiplied by 
100, the square of the length, and by 7, the horizontal distance 
between bearings, and by five-eighths of the weight, 12*5, 
equals 21,875. The greatest value of n^ Table IL, for hem- 
lock, 0-08794, multiplied by 1240, the value of E, Table IL, 
for hemlock, equals 109-0456. The above 21,875, divided by 
109-0456, equals 200*6, the value of U. I^ow by Rule 
XXXU., the above 200-6, divided by 0*589, equals 340*6, the 
square root of which is 18*46, and the square root of this 
square root is 4*296, the diameter required — equal to 4/^ 
inches nearlj'. 

329. — The greater the depth of a beam in proportion to the 
thickness, the greater the strength. But when the difference 
between the depth and the breadth is great, the beam must be 
stayed, (as at Fig. 228,) to prevent its falling over and break- 
ing sideways. Their shrinking is another objection to deep 
beams ; but where these evils can be remedied, the advantage 
of increasing the depth is considerable. The following rule is, 
to find the strongest form for a heam out of a given quantity 
of timher. 

Rule. — Multiply the length in feet by the decimal, 0*6, and 
divide the given area in inches by the product ; and the 
square of the quotient will give the depth in inches. 

Examjple. — What is the strongest form for a beam whose 
given area of section is 48 inches, and length of bearing 20 
feet ? The length in feet, 20, multiplied by the decimal, 0*6, 
gives 12 ; the given area in inches, 48, divided by 12, gives a 
quotient of 4, the square of which is 16 — this is the depth in 
inches ; and the breadth must be 3 inches. A beam 16 inches 

28 



218 



AMERICAN HOUSE-CAKPENTEE. 



by 3 would bear twice as mucb as a square beam of the same 
area of section ; which shows how important it is to make 
beams deep and thin. In many old buildings, and even in 
new ones, in country places, the very reverse of this has been 
practised ; the principal beams being oftener laid on the 
broad side than on the narrower one. 

The foregoing rules, stated algebraically, are placed in the 
following table. 

TABLE V. STIFFNESS OF BEAMS I DIMENSIONS. 



When 

the 

beam 

is laid 


When the weight is 


Eectangular. 


Square. 


Bound. 


Value of 
depth. 


Value of 
breadth. 


When &=(?/-, 
value of d. 


Value of a 
side. 


Value of the 
diameter. 




Concentrated at middle 
Equally distributed 
By the foot superficial 


(62.) 
Enh 


(63.) 

wl- 

End^ 


(64.) 
Enr 


(65.) 
i/wP 

^ En 


(66.) 
^ -589 En 


1 
1 


(67.) 
En b 


(68.) 
End^ 


(69.) 
Enr 


(70.) 
^ En 


(71.) 
^ QmEn 




(72.) 
Enb 


(73.) 
End^ 


(74.) 
Enr 


(75.) 
En 


(76.) 
^•9424^71 




Concentrated at middle 
Equally distributed 
By the foot superficial 


(77.) 
^ Enh 


(78.) 
wlh 
Endi 


(79.) 
Enr 


(80.) 


(81.) 
^'5S9 En 


fee 

a 
"a 

=5 

a 


(82.) 
^ Enb 


(83.) 
End^ 


(84.) 
Enr 


(85.) 
^—ElT 


(86.) 
*/ wlh 
•94M En 




(S7.) 
Enb 


(88.) 
%.fcl-h 
Endi 


(89.) 
Enr 


(90.) 


(91.) 
^•9424^71 



In the above table, I equals the breadth, and d the depth of 
cross-section of beam ; oi equals the rate of deflection per foot of 
the length ; 5, d and w, all in inches. Also, I equals the length, 
c the distance between two parallel beams measured from the 



FEAMING. 



219 



centres of their breadth, and A equals the horizontal distance 
between the supports of an inclined beam ; Z, c and A, all in 
feet. Again, lo equals the weight on the beam,/* equals the 
weight upon each superficial foot of a floor or roof, supported 
by two or more beams laid parallel and at equal distances 
apart ; w and f in pounds. And r is any decimal, chosen at 
pleasure, in proportion to unity, as & is to d — from which pro- 
portion h — dr. ^ is a constant the value of which is found 
in Table 11. 

330. — To ascertain the scantling of the stiffest team that can 
le cut from a cylinder. Let d a ch^ {Fig. 223,) be the section, 
and e the centre, of a given cylinder. Draw the diameter, 
ah J upon a and J, with the radius of the section, describe tlie 
arcs, d e and e c ; join d and a^ a and c, c and 5, and I and d ; 
then the rectangle, d a g h^ will be a section of the beam 
required. 




Fig. 223. 



331. — Resistance to Rupture. — ^The resistance to deflect/lo^i 
having been treated of in the preceding articles, it now re- 
mains to speak of the other branch of resistance to cross 
strains, namely, the resistance to rupture. When a beam is 
laid horizontally and supported at each end, its strength to resist 
a cross strain, caused by a weight or vertical pressure at the 
middle of its length, is directly as the breadth and square of 
the depth and inversely as the length. If the beam is square, 
or the depth equal to the breadth, then the strength is directly 



220 A3IEEICAX ] [OUSE-CAEPEXTEK. 

as the cube of a side of the beam and iuversely as tlie length, 
and if the beam is round the strength is directly as the cube 
of the d^'ameter and inversely as the length. 

When the weight is concentrated at any point in the length, 
the strength of the beam is directly as the length, breadth, and 
square of the depth, and inversely as the product of the two 
parts into which the length is divided by the point at which 
the weight is located. 

TVhen the beam is laid not horizontal but inclining, the 
strength is the same as in each case above stated, and also in 
proportion, inversely as the cosine of the angle of inclination 
with the horizon, or, which is the same thing, directly as the 
length and inversely as the horizontal distance between the 
points of support. 

When the weight is equally diffused over the length of a 
beam, it will sustain just twice the weight that could be sus- 
tained at the middle of its length. 

A beam secured at one end only, will sustain at the other 
end just one-quarter of the weight that could be sustained at 
its middle were the beam supported at eaeh end. 

These relations between the strain and the strength exist in 
all materials. For any particular kind of material, 

;4 = S; (92.) 

S, representing a constant quantity for all materials of like 
strength. The superior strength of one kind of material over 
another is ascertained by experiment ; the value of S being 
ascertained by a substitution of the dimensions of the piece tried 
for the symbols in the above formula. Having thus obtained 
the value of S, the formula, by proper inversion, becomes use- 
ful in ascertaining the dimensions of a beam that wiU require 
a certain weight to break it ; or to ascertain the weight that 
will be required to break a certain beam. It will be observed 
in the preceding formula, that if each of the dimensions of the 



FRAMING. 



221 



beam equal unity, then w = S. Hence, S is equal to the 
weight required to break a beam one inch square and one 
foot long. The values of S, for various materials, have been 
ascertained from experiment, and are here recorded : — 



TABLE VI. STEENGTH. 



Materials. 


Value of ^. 


Number of 
Experiments. 


Green plate-glass 

Spruce . 

Hemlock 

Soft white pine 

Hard white pine 

Ohio yellow pine 

Chestnut 

Georgia pine 

Oak 

Locust 

Cast-iron (from 1550 to 2280) .... 


178 
845 
863 
890 
449 
454 
503 
510 
5T4 
742 
1926 


4 
5 
7 
9 
1 
2 
2 
7 
2 
2 
29 



The specimens broken were of various dimensions, from one 
foot long to three feet, and from one inch square to one by 
three inches. The cast-iron specimens were of the various 
kinds of iron used in this country in the mechanic arts. S 
may be taken at 2,000 for a good quality of cast-iron. It is 
usual in determining the dimensions of a beam to suppose it 
capable of sustaining safely one-third of the breaking weight, 
and yet Tredgold asserts that one-fifth of tie breaking weight 
will in time injure the beam so as to give it a permanent set 
or bend, and Hodgkinson says that cast-iron is injured by any 
weight however small, or, in other words, that it has no elastic 
power. However this may be, experience has proved cast- 
iron quite reliable in sustaining safely immense weights for 
a long period. Practice has shown that beams will sustain 
safely from one-third to one-sixth of their breaking weight. 
If the load is bid on quietly, and is to remain wdiere laid, at 
rest, beams may be trusted with one-third of their breaking 
weight, but if the load is moveable, or subject to vibration, 



222 AMERICAN HOUSE-CAEPENTER 

one-quarter, one-fifth, or even, in some cases, one-sixth is quite 
a sufiicient proportion of the breaking load. 

332. — ^The dimensions of beams should be ascertained only 
by means of the rules for the stiffness of materials, (Arts. 320 
323, et seq.,) as these rules show more accurately the amount 
of pressure the material is capable of sustaining without injury. 
Yet owing to the fact that the rules for the strength of mate- 
rials are somewhat shorter, they are more frequently used 
than those for the stiffness of materials. In order that the 
2?roportion of the breaking weight may be adjusted to suit cir- 
cumstances it is well to introduce into tlie formula a symbol 
to represent it. The proportion represented by the symbol 
may then be varied at discretion. Let this symbol be a, a 
decimal in proportion to unity as the safe load is to the break- 
ing load, then S a will equal the safe load. Hence, 

S ah d' ,^o \ 

w = J (93.) 

for a safe load at middle on a horizontal beam supported at 
both ends ; and 

w = J (94.) 

for a safe load equally diffused over the length of the beam ; 

and 

^ ^ S ah d^ ,^^ . 

f=—jj- (95.) 

for the load, per superficial foot, that can be sustained safely 
upon a floor supported by two or more beams, c being the dis- 
tance in feet from centres between each two beams, and f the 
load in pounds per superficial foot of the floor. Generally, in 
(93,) (94,) and (95,) w equals the load in pounds ; aS, a constant, 
the value of which is found in Table YI. ; a a decimal, in pro- 
portion to unity as the safe load is to the breaking load ; I the 
length in feet between the bearings ; and h and d the breadth 
and depth in inches. 



FRAMING. 



22; 



TO FIND THE WEIGH f. 

333. — ^The formulas for ascertaining the weight in the seve- 
ral cases are arranged in the following table, where c, /, w, S^ 
a, ly h and d represent as above ; and also s equals a side of 
a square beam ; D equals the diameter of a cylindrical beam ; 
m and n equal respectively the two parts into which the length 
is divided by the point at which the weight is located ; and A 
equals the horizontal distance between the supports of an 
inclined beam. 



TABLE Vn. STRENGTH OF BEAMS ; SAFE WEIGHT. 



"When the 


When the weight is 


When the beam is 


beam is laid 


Eectangular. 


Square. 


Eonnd. 




Concentrated at middle, lo, in 
pounds, equals 

Equally distrihuted, w, in 
pounds, equals 

By the foot superficial, / in 
pounds, equals 

Concentrated at any point in 
the length, w, in pounds, equals 


(96.) 
Sahd^ 


(9T.) 
I 


(98.) 
•589 D^S a 




I 


I 


• 1 


(99.) 

2;Sa6c?2 

I 


(100.) 
I 


(101.) 
1-178 If^S a 


(102.) 

C ti 


a03.) 

2^a.-?3 

cl- 


(104) 

1-178 D^S a 

cP 




(105.) 

Sahd-^l 

A.mn 


(106.) 
4: mil 


(107.) 

'U7DiSal 

m 11 




Concentrated at middle, w, in 
pounds, equals 

Equally <iistrihuted, w, in 
ponnds, equals 

By the foot superficial, ;; in 
pounds, equals 


(108.) 

Sahdr- 

h 


(109.) 

Sas^ 

h 


(110.) 

■589D'Sa 

h 


1 
1 


(111.) 
'iSa'bd'i 
h - 


(112.) 
Ji 


(113.) 
1178 D^S a 

h 


(114.) 
2SaJ>d-i 
■ chl ' 


(115.) 
2^rts' 

~nrr 


(116.) 

1-178 D^S a 

chl 




Concentrated at any point in 
the length, w, in pounds, equals 


(117.) 

Sahdi P 

4hmn 


(118.) 
4kmn 


(119.) 
147 D^SaP 
hmn - 



224: AMERICAN H0U6E-CARPENTEE. 

Practical Mules and Examples. 

Mule XXXIII. — To find the weight that may be supported 
safely at the miiddle of a beam laid horizontally. Multiply 
the value of 8^ Table YI., by a decimal that is in proportion 
to unity as the safe weight is to the breaking weight, and 
divide the product by the length in feet. Then, if the beam 
is rectangular, multiply this quotient by the breadth and by 
the square of the depth, and the product will be the required 
weight in pounds ; or, if the beam is square, multiply the said 
quotient, instead, by the cube of a side of the beam and the 
product will be the required weight in pounds ; but, if the 
beam is round, multiply the aforesaid quotient, instead, by 
•589 times the cube of the diameter, and the product will be 
the required weight in pounds. 

Exam.ple. — ^What weight will a rectangidar white pine 
beam, 20 feet long, and 3 by 10 inches, sustain safely at the 
middle, the portion of the breaking weight allowable being 
0*3? By the above rule, 390, the value of .iS' for white pine. 
Table YL, multiplied by 0*3, the decimal referred to, equals 
117, aPid this divided by 20, the length, the quotient is 5-85. 
Now the beam being rectangular, this quotient multiplied by 
3 and by 100, the breadtli and tlie square of the depth, the 
product, 1755, is the desired weight in pounds. 

Example. — If the above beam had been square^ and 6 by 6 
inches, then the quotient, 5'85, multiplied by 216, the cube of 
6, a side, the product, 1263*6, is the weight required in pounds. 

Example. — If the above beam had been rounds and 6 inches 
diameter, then the above quotient, 5*85, multiplied by '589 
times 216, the cube of the diameter, the product, 744*26, would 
be the required weight in pounds. 

Mule XXXIY. — To find the weight that may be sujpported 
safely when equally distributed over the length of a beam, 
lai(^ horizontally. Multiply the result obtained, by Eule 



FRAMING. ' 225 

XXXni., bj 2, and the product will be the required weight 
in pounds. 

Example, — In the example, under Rule XXXIIL, the safe 
weight at middle of rectangvlar beam is found to be 1755 
pounds. This multiplied by 2, the product, 3510, is the weight 
the beam will bear safely if equally distributed over its length. 

Exanvple. — So in the case of the square beam, 2527*2 pounds 
is the weight, equally distributed, that may be safely sus- 
tained. 

Example. — And for the round beam 1488*52 is the required 
weight. 

Bule XXXY. — To ascertain the weight per superficial foot 
that may be safely sustained on a floor resting on two or more 
beams laid hm^izontally and parallel. Multiply twice the value 
of aS", Table YL, by the decimal that is in proportion to unity, 
as the safe weight is to the breaking weight, and divide tl e 
product by the square of the length, in feet, multiplied by the 
distance apart, in feet, between the beams measured from their 
centres. E'ow, if the beams are rectangular^ multiply this 
quotient by the breadth and by the square of the depth, both 
in inches, and the product will be the required weight in 
pounds ; or if the beams are square^ multiply said quotient, 
instead, by the cube of a side of a beam and the product will, 
be the required weight in pounds. But if the beams are 
rounds multiply the aforesaid quotient, instead, by '589 times 
the cube of the diameter, and the product will be the weight 
required in pounds. 

Example. — What weight may be safel}' sustained on each 
foot superficial of a floor resting on spruce beams, 10 feet long,. 
3 by 9 inches, placed 16^ inches, or li feet, from centres: the 
portion of the breaking weight allowable being 0*25? By the 
Rule, 690, twice the value of spruce, Table YI., multiplied by 
0*25, the decimal aforesaid, equals 172*5. This product divided 
by 100, the square of the length, multiplied by 1|-, the distance 

29 



226 AMEKICAIT HOUSE-CARPENTEE. 

from centres, equals 1-294. Kow tliis quotient multiplied by 
3j the breadth, and bv 81, the square of the depth, the product 
314*44: is the required weight in pounds. 

Had these beams been square^ and 6 bj 6 inches, the re- 
quired weight would be 279*5 pounds. 

Or, a round, and 6 inches diameter, 164*63 pounds. 

Itule XXXYI. — To ascertain the weight that may be sus- 
tained safely on a beam when concentrated at any point of its 
length. Multiply the value of S, Table YL, by the decimal in 
proportion to unity, as the safe weight is to the breaking 
weight, and by the length in feet, and divide the product by 
four times the product of the two parts, in feet, into which the 
length is divided, by the point at which the weight is concen- 
trated. Then, if the beam is rectangular^ multiply this quo- 
tient by the breadth and by the cube of the depth, both in 
inches, and the product will be the required weight in pounds. 
Or, if the beam is square^ multiply the said quotient, instead, 
b}^ the cube of a side of the beam, and the product w31 be the 
required weight in pounds. But if the beam is rounds multi- 
ply the aforesaid quotient by '589 times the cube of the dia- 
meter, and the product will be the weight required. 

Example. — "What weight may be safely supported on a 
Oeorgia pine beam, 5 by 12 inches, and 20 feet long ; the weiglit 
pla<?ed at 5 feet from one end, and the proportion of the break- 
ing weight allowable being 0*2 ? By the rule, 510, the value of 
S for Georgia pine. Table YI., multiplied by 0*2, the decimal 
referred to, equals 102 ; this by 20, the length, equals 2040 ; 
this divided by 300, (= 4 X 5 x 15,) or 4 times the product of 
the two parts into which the length is divided by the point at 
which the weight is located, equals 6*8. The beam being rect- 
angular, this quotient multiplied by 5, the breadtli, and by 
144, the square of the depth, equals 4896, the required weight. 

A beam, 8 inches sq_iiare, other conditions being the same as 
in the preceding case, would sustain safely 3481*6 pounds. 



FRAMING. 22f 

And a round beam, 8 inches diameter, will sustain safely, 
under like conditions, 2050*66 pounds. 

Rule XXXYII.— To find the weight that may be safely 
sustained on inclined beams. Multiply the result found for 
horizontal beams in preceding rules, by the length, in feet, 
and divide the product by the horizontal distance between the 
supports, in feet, and the quotient will be the required weight. 

Example. — What weight may be safely sustained at the 
middle of an oak beam, 6 X 10 inches, and 10 feet long, (set 
inclining, so that the horizontal distance between the supports 
is 8 feet,) the portion of the breaking weight allowable being 
0*3 ? The result for a horizontal beam, by Eule XXXIIL, is 
10,332 pounds. This, multiplied by 10, the length, and divid- 
ed by 8, the horizontal distance, equals 12,915 pounds, the 
required weight. 

TO FIND THE DIMENSIONS. 

334. — ^The following table exhibits, algebraically, rules for 
ascertaining the dimensions of beams required to support 
given weights ; where h equals the breadth, and d the depth 
of a rectangular beam, in inches ; I the length between sup- 
ports ; h the horizontal distance between the supports of an 
inclined beam, and c the distance apart of two parallel beams, 
measured from the centres of their breadth, ?, \ and c, in feet; 
w equals the weight on a beam ; f the weight on each super- 
ficial foot of a floor resting on two or more parallel beams ; B, 
equals a load on a beam, and m and n the distances, respect- 
ively, at which R is located from the two supports ; also P is 
a weight, and g and Tc the distances, respectively, at which P 
is located from the two supports ; also m-^-n^l^g + hi w, 
/*, i?, and 7-*, all in pounds ; m, n, g, and ^, in feet. /S' is a 
constant, the value of which is found in Table YL ; « is a 
decimal in proportion to unity as the safe load is to the break- 



228 



AMEPJCA^' HOUSE-CAEPEXrZE. 



ing load; r is a decimal in proportion to nnitv as 5 is to d ; 
from which h — dr ; a and t to be chosen at discretion. 



TABLE Tin. STRENGTH 



When the 


When the 
weight is 


Eectangular. 


beam ia laid 


Yalne of depth. 


Value of breadth. 




Concentrated at 
middle 

Equally distribut- 
ed 

By tbe foot snper- 

CoDMntrated at 
any point in the 
length 

At tn-o or more 
points in the 
length 


(120.) 


(121.) 
Sadi 




(125.) 
2 Sal 


(126.) 

tcl 

TSad^- 


1 


(180.) 
28ab 


(131) 

fc I* 

2 Sad^ 




(135.) 
Sabl 


(136.) 
4:v:mn 
Sad^l 






(141) 
A{Bmn^Pgk^ rfc.) 




Sahl ' 


Sadil 




Concentrated at 
middle 

BqoflDy distribnt- 
ed 

By the foot super- 
ficial 

Concentrated at 
any point in the 

At two or more 
points in the 
leng^th 


a45.) 
Sab 


(146.) 
Sad^ 




(150.) 
^ tch 
2Sab 


(151) 
2 5a<f' 


1 

1 


(155.^ 
2Sab 


(156.) 
fehl 
2Sadt 




(160.) 
~S^bP 


(161.) 
4Atf mn 




ass.) 

V-t A {Bmn-hPgk-i- «fej.) 
SabP 


a66.) 

4»(5mn^i'^t + «fej.) 



FRAMING. 



229 



OF BEAMS ; DIMENSIONS. 





Square. 


Eound. 


When J> ^dr. value of d. 


Value of a side. 


Yalue of the diameter. 


(122.) 
Sar 


(123.) 

l/wl 

Sa 


(124.) 

'-msa 


(127.) 
3/ wl 
2Sar 


(128.) 
l/wl 


(129.) 
"^ 1-178^ a 


(132.) 
2Sar 


(133.) 
'^2^a 


(134.) 
1-178 S a 


(137.) 
^ Sari 


(138.) 

3/4 40 TO 71 


(139.) 
^•147^aZ 


(142.) 
^ Sari 


(143.) 
Sal 


(144.) 

»/(i?TO» + i>^^ + <ec.) 

•147^ai 


(147.) 
l/wh 
Sar 


(148.) 


(149.) 
3/ wh 

•5S9 5a 


(152.) 
2^ar 


(153.) 
l/wh 
2Sa 


(154.) 
3/ «,A 

1-178 /S a 


(157.) 
l//ehl 
2 Sar 


(158.) 
"^ 2^a 


(159.) 
1-178 .S a 


(162.) 
Sarir 


(163.) 
8/4 A -j/j TO n 


(164.) 
^ hwrnn 


(167.) 
»/4A(JJ»»»t + P^fc + «fec.) 


(168.) 

i/47i(Ilmn + Pgk + dbc.) 

Sal. 


(169.) 

«/^ (Jimn + Fffk + <fcc.) 

-147 ^a^ 



230 AMEEICAN HOUSE-CAEPENTEE. 

PraGtical Rules and Examj^les, 

Rule XXXYIII. — Preliminary. When the weight is con- 
centrated at the middle. Multiply the weight, in pounds, by 
the length, in feet, and divide the product by the value of S^ 
Table YL, multiplied by a decimal that is in proportion to 
unity as the safe weight is to the breaking weight, and the 
quotient is a quantity which may be represented by «/, referred 
to in succeeding rules. 

^' = ^ (m.) 

Rule XXXIX. — Preliminar3^ Wlien the weight is equally 
distributed. One-half of the quotient obtained by the preced- 
ing rule is a quantity which may be represented by K^ referred 
to in succeeding rules. 

Ricle XL. — Preliminary. When the weight is per foot su- 
perficial. Multiply the weight per foot superficial, in pounds, 
by the square of the length, in feet, and by the distance apart 
from centres between two parallel beams, and divide the pro- 
duct by twice the value of S^ Table YL, multiplied by a deci- 
mal in proportion to unity as the safe weight is to the break- 
ing weight, and the quotient is a quantity which may be re- 
presented by Z, referred to in succeeding rules. 

Ride XLI. — Preliminary. When the weight is concentrated 
at any point in the length. Multiply the distance, in feet, 
from the loaded point to one support, by the distance, in feet, 
from the same point to the other support, and by four times 
the weight in pounds, and divide the product by the value of 
S^ Table YL, multiplied by a decimal in proportion to unity 
as the safe weight is to the breaking weight, and by the length, 



FRA:NnNCr. 231 

in feet ; and the quotient is a quantity wliicli may be repre- 
sented by Q^ referred to in tlie rules. 

Rule XLII. — Preliminary. When two or more weights are 
concentrated at any joints in the length of the 'beams. Multi- 
ply each weight by each of the two parts, in feet, into w^hich 
the length is divided by the point at which the weight is 
located, and divide four times the sum of these products by 
the value of S^ Table YL, multiplied by a decimal in propor- 
tion to unit}^ as the safe weight is to the breaking weight, and 
by the length, in feet, and the quotient is a quantity which 
may be represented by F, referred to in the rules. 
4:{Emn + P glc-\-<&G. ) ^ y 
lo a 

Mule XLIII. — Preliminary. When the learn is not laia 
horizontal^ hut inclining. In the five preceding preliminary 
rales, multiply the result there obtained by the horizontal dis- 
tance between the supports, in feet, and divide the product by 
the length, in feet, and the quotient in each case is to be used 
for beams when inclined, as referred to in succeeding rules. 



TO FIND THE DIMENSIONS. 

Rule XLIY. — When the beam is rectangular. To find the 
depth. Divide the quantity represented in preceding rules by 
f/, K^ Z, Q^ or F", by the breadth, in inches, and the square 
root of the quotient will be the depth required in inches. 

To find the hreadth. Divide the quantity represented by J, 
K^ Z, Q, or F, by the square of the depth, and the quotient 
will be the required breadth, in inches. 

To find both Ireadth and dejpth^ when they are to be in a 
given proportion. Divide the quantity represented by J, K, 
Z, §, or F, by a decimal in proportion to unity as the breadth 



232 AMERICAN HOUSE-CAEPENTEE. 

IS to be to the depth, and the cube root of the quotient ^*ill be 
the depth in inches. Multiply the depth by the aforesaid de- 
cimal and the quotient will be the hreadt?. in inches. 

Example. — A locust beam, 10 feet long in the clear of the 
supports, is required to sustain safely 3,000 pounds at the 
middle of its length, the portion of the breaking weight allow- 
able being 0*3 ; what is the required breadth and depth ? 
Proceeding by the rule for weight concentrated at middle, 
(Rule XXXYIII.,) 3,000, the weight, by 10, the length, equals 
30,000. The value of S, Table YL, for locust, is 742 : this by 
0*3 the decimal, as above, equals 222*6 ; the 30,000 aforesaid 
divided by this 222*6 equals 134*77, equals the quantity repre- 
sented by J. Now to find the depth when the breadth is 4 
inches, 134*77 divided by 4, the breadth, as above required, 
the quotient is 33*69, and the square root of this, 5*8, is the 
required depth in inches. But to find the breadth, when the 
depth is known, let the depth be 6 inches, then 134*77 
divided by 36, the square of the depth, equals 3*74, the breadth 
required in inches. Again, to find both breadth and depth in 
a given proportion, say, as 0*6 is to 1*0. Here 134*77 divided 
by 0*6 equals 224*617, the cube root of which is 6*08, the re- 
quired depth in inches, and 6*08 by 0*6 equals 3*648, the re- 
quired breadth in inches. 

Thus it is seen, in this example, that a piece of locust tim- 
ber, 10 feet long, having 3,000 pounds concentrated at the 
middle of its length, as tV of its breaking load, is required to 
be 4 by 5f^ inches, or 3f by 6 inches, or 3^ by Q^ inches. If 
this load were equally diffused over the length, the dimensions 
required would be found to be 4 by 4*1, or 1*87 by 6, or 2*895 
t)y 4*825 inches, in the three cases respectively. 

Example. — A tier of chestnut beams, 20 feet long, placed 
cue foot apart from centres, is required to sustain 100 pounds 
per superficial foot upon the floor laid upon them : this load 
to be 0*2 of the breaking weight ; what is the required dimen- 



FRAMING. 233 

sions of the cross-section ? By Kule XL., the rule for a load 
per foot superficial, 100 by 20 x 20 and by 1 equals 40,000. 
Twice 503, the value of jS for chestnut. Table YI., by 0-2 
equals 201-2. The above 40,000 divided by 201-2 equals 198*8, 
the value of Z. Kow if the breadth is known, and is 3 inches, 
198*8 divided by 3 equals 66-27, the square root of which is 
8-14, the required depth. But if the depth is known and is 9 
inches, 198-8 divided by (9x9==) 81 equals 2*454 inches, the 
required breadth. Again, when the breadth and depth are 
required in the proportion of 0-25 to 1*0, then 198*8 divided 
by 0*25 equals 795-2, the cube root of which is 9*265, the 
required depth in inches, and 9-265 by 0*25 equals 2*316, the 
required breadth in inches. 

Examjple. — A cast-iron bar, 10 feet long, is required to sus- 
tain safely 5,000 pounds placed at 3 feet from one end, and 
consequently at 7 feet from the other end, the portion of the 
breaking load allowable being 0*3 ; what must be the size of 
the cross section ? By Rule XLL, the rule for a concentrated 
load at any point in the length of the beam, 3 x 7 X 4 x 5000 
= 420,000. And 1926, the value of 8 for cast-iron. Table YL, 
by 0-3 and by 10 equals 5778. The aforesaid 420,000 divided 
by this, 5778, equals 72-689, the value of Q. l^ow if the 
breadth is fixed at 1-5 then 72-689 divided by 1*5 eqwals 
48-459, the square root of which is 6*96, the required depth in 
inches. But if the depth is fixed at 6 inches, then 72*689, the 
value of Q^ divided by 36, the square of 6, equals 2*019, the 
required breadth in inches. Again, if the breadth and depth 
are required in the proportion 0-2 to 1-0 ; then Q^ 72-689, 
divided by 0*2, equals 363*445, the cube root of which is 7*136, 
the required depth in inches; and 7*136 by 0*2 equals 1*427, 
the required breadth in inches. 

liiile XLY. — "When the beam is square to find the breadth 
of a side. The cube root of the quantity represented by J^ K^ 
Z, Q^ or F, in preceding rules, is the breadth of the side required 

30 



234: AMERICAN HOUSE- GAEPENTEE. 

Example. — A Georgia pine, beam, 10 feet long, is reqiured 
to sustain, as 0-3 of the breaking load, a weight of 30,000 
pounds equally distributed over its length, and the beam to be 
square, what must be the breadth of the side of such a beam ? 
By the rule for an equally distributed load, (Kule XXXIX.,) 
30,000 X 10 = 300,000, and 510 (the value of S, for Georgia 
pine, Table YL) x 0*3 = 153. 300,000 divided by 153 equals 
1960-Y84, and one-half of this equals 980-392, the value of K 
N^ow the cube root of this is 9*931 inches, or 9-|-|, the required 
side. Had the weight been concentrated at the middle 
1960-784: would be the value of «/, and the cube root of this 
12-515, or 12J inches, would be the size of a side of the 
beam. 

Example. — A square oak beam, 20 feet long, is required to 
sustain, as 0-25 of the breaking strength, three loads, one of 
8,000 pounds at 5 feet from one end, one of 7,000 pounds at 
14 feet, and one of 5,000 pounds at 8 feet from one end, what 
must be the breadth of a side of the beam ? The value of S^ 
for oak, Table YI., is 574. By the rule for this case, (Kule 
XLII.,) 8000 X 5 X 15 equals 600,000 ; and 7000 X 14 x 6 
equals 588,000 ; and 5000 x 8 x 12 equals 480,000. The sum 
of these products is 1,668,000 ; this by 4 equals 6,672,000. 
Now 574 X 0-25 x 20 equals 2870, and the 6,672,000 divided 
by the 2870 equals 2325, the number represented by F/ the 
cube root of which is 13-25, the required size of a side of the 
beam, 13J inches. This is for a horizontal beam. ITow if 
this beam be laid inclining, so that the horizontal distance 
between the bearings is 15 feet, then to find the size by the 
rule for this case, viz. XLIJI., the above number "F, equal to 
2325, multiplied by 15, the horizontal distance, equals 34,875, 
and this divided by 20, the length, equals 1743*75. ISTow by 
E-ule XLY., the cube root of this is 12*04, the required size of 
a side — 12 inches full. 

Bule XLYI. — When the beam is round. Divide the quan- 



FRAMING. 235 

tity represented by «/, K^ Z, Q^ or Tby the decimal 0-589, and 
the cube root of the quotient will be the required diameter. 

Example. — A white pine beam or pole, 10 feet long, is re- 
quired to sustain, as the 0*2 of the breaking strength, a load 
of 5,000 pounds concentrated at the middle, what must be the 
diameter ? The value of S, for white pine, Table YI., is 390. 
Now by the rule for load at middle, (XXXVIII.,) 5000 x 10 
= 50,000 ; and 390 x 0-2 = 78 ; and 50,000 ^ 78 = 641 = J. 
By this rule, 641 -f- 0-589 = 1088-28, the cube root of which, 
10-28, is the required diameter. If this beam be inclined, 
so that the horizontal distance between the supports is 7 feet, 
then to find the diameter, by Kule XLIII., value of J as 
above, 641 multipUed by 7 and divided by 10 equals 448-7. 
Now by this rule, 448-7 -^ 0-589 = 761-796, the cube root of 
which, 9*133, is the required diameter. 

.JExample. — A spruce pole, 10 feet long, is required to sus- 
tain, as the 0-33i or -J of the breaking weight, a load of 1,000 
pounds at 3 feet from one end, what must be the diameter ? 
The value of S for spruce. Table YI., is 345. By the rule for 
this case, (Rule XLI.,) 3 x 7 x 4 x 1000 =. 84,000 ; and 345 
X i X 10 = 1150 ; and 84,000 -^ 1150 = 93-04, the value of Q. 
Now by this rule, 73-04 -^ -589 = 124, the cube root of which, 
4-9866, is the required diameter in inches. 

335. — Systems of Framing. In the various parts of framing 
known as floors, partitions, roofs, bridges, &c., each has a spe- 
cific object ; and, in all designs for such constructions, this 
object should be kept clearly in view ; the various parts being 
so disposed as to serve the design with the least quantity of 
material. The simplest form is the best, not only because it is 
the most economical, but for many other reasons. The great 
number of joints, in a complex design, render the construction 
liable to derangement by multiplied compressions, shrinkage, 
and, in consequence, highly increased oblique strains ; by 
which its stability and durabiUty are greatly lessened. 



236 



AMEPJCAI3 HOUSE-CARPENTEE. 



FLOORS. 



336. — Floors aye most generally constructed single, that is^ 
eimply a series of parallel beanas, eacli spanning the width ol 




Fig 224. 



the floor, as seen at Fig, 224. Occasionally floors are con- 




Fig. 225. 



structed double, as at Fig. 225 ; and sometimes framed, as at 



FRAMESTG. 



237 




Fig. 226. 



Fig. 226 ; but these methods are seldom practised, inasmuch 
as either of these require more timber than the single floor. 
Where lathing and plastering is attached to the floor beams to 
form, a ceiling below, the springing of the beams, by custo- 
mary use, is liable to crack the plastering. To obviate this in 
good dwellings, the double and framed floors have been 
resorted to, but more in former times than now, as the Gross- 
furring (a series of narrow strips of board or plank, nailed 
transversely to the underside of the beams to receive the lath- 
ing for the plastering,) serves a like purpose very nearly as well. 
337. — In single floors the dimensions of the beams are to be 
ascertained by the preceding rules for the stiffness of materials. 
These rules give the required dimensions for the various kinds 
of material in common use. The rules may be somewhat 
abridged for ordinary use, if some of the quantities repre- 
sented in the formula be made constant within certain limits. 
For example, if the load per foot superficial, and the rate of 
deflection, be fixed, then these, together with the f, and the 



238 AMERICAN HOUSE-CAEPENTEE. 

constant represented by E^ may be reduced to one ccmstant. 
For dwellings, tlie load per foot may be taken at ^^ pounds, 
as this is the weight, that has been ascertained by experiment, 
to arise from a crowd of people on then- feet. To this add 20 
for the weight of the material of which the floor is composed, 
and the sum, 86, is the value of /", or the weight per foot 
superficial for dwellings. The rate of deflection allowable for 
this load may be fixed at 0*03 inch per foot of the length. 
Then (M) transposed. 



becomes 



F^.=^^' 



5 X 86 cl' _.^ 
8 X 0-03 X ^ - ^ ^ 



which, reduced, is 

'^XGV = ld' (175.) 

Kediicing — -p- for five of the most common woods, and there 

results, rejecting small decimals, and putting —p~ = a?, a; 
equal, fo:- 

Georgia pine 0'6 

Oak 0-7 

White pine I'O 

Spruce 1*15 

Hemlock 1*45 

Therefore, the rule is reduced to x cV — h d\ And for 
lohite pine^ the wood most nsed for floor beams, a? = 1*0, and 
therefore disappears from the formula, rendering it still more 
simple, thus, 

cr = ld' (176.) 

The dimensions of beams for stores, for all ordinary business, 
may also be calculated by this modified rule, (175,) for it will 
require about 3^ times the weight used in this rule, or about 



FRAMING. 239 

300 pounds, to increase the deflection to the limit of elasticity 
in white pine, and nearly that in the other woods. But for 
warehouses, taking the rate of deflection at its limit, and fixing 
the weight per foot at 500 pounds, including the weight of the 
material of which the floor is constructed, and letting y repre- 
sent the constant, then 

ycV^ld' (177.) 

and y equals, for 

Georgia pine . . . . . , . 1-35 

Oak 1-35 

White pine 1"75 

Spruce 2-2 

Hemlock 2-85 

338. — Hence to find the dimensions of floor beams for dvjell- 
ings when the rate of deflection is 0*03 inch per foot, or for 
ordinary stores when the load is about 300 pounds per foot, 
and the deflection caused by this weight is within the limits 
of the elasticity of the material, we have the following rule : 

Bide XLYII. — Multiply the cube of the length by the dis- 
tance apart between the beams, (from centres,) both in feet, 
and multiply the product by the value of a?, {Art. 337.) E'ow 
to find the breadth, divide this product by the cube of the 
depth in inches, and the quotient will be the breadth in inches. 
But if the depth is sought, divide the said product by the 
breadth in inches, and the cube root of the quotient will be 
the depth in inches ; or, if the breadth and deptli are to be in 
proportion, as r is to unity, r representing any required deci- 
mal, then divide the aforesaid product by the value of t*, and 
extract the square. root of the quotient, and the square root of 
tliis square root will be the depth required in inches, and the 
depth multiplied by the value of r will be the breadth in 
inches. 

Example. — To find the Ireadth. In a dwelling or ordinary 
store what must be the breadth of the beams, when placed 15 



240 AMEEICA25- HOUSE-CARPENTER. 

inches from centres, to support a floor covering a span of 16 
feet, the depth being 11 inches, the beams of oak ? Bj the 
rule, 4096, the cube of the length, by H, the distance from 
centres, and by O'T, the value of a?, for oak, equals 3584. This 
divided bv 1331, the cube of the depth, equals 2*69 inches, or 
2|i inches, the required breadth. 

JExample. — To find the depth. The conditions being the 
same as in the last example, what must be the depth when the 
breadth is 3 inches. The product, 3584, as above, divided by 
3, the breadth, equals 1194f ; the cube root of this is 10*61, or 
lOf inches nearly. 

Example. — To find the hreadth and depth in proportion.^ 
say, as 0*3 to 1*0. The aforesaid product, 3584, divided by 
0*3, the value of r, equals ll,946f , the square root of which is 
109*3, and the square root of this is 10*45, the required depth. 
This multiplied by 0*3, the value of r^ eqnals 3*135, the re- 
quired breadth, the beam is therefore to be 3-|- by 10-| inches. 

339. — And to find the breadth and depth of the beams for a 
floor of a vjarehouse sufficient to sustain 500 pounds per foot 
superficial, (including weight of the material in tlie floor,) with 
a deflection not exceeding the limits of the elasticity of the 
material, we have the following rule : 

Hide XL V 111. — The same as XLYII., with the exception 
that the value of y {Art. 337) is to be used instead of the 
value of X. 

Example. — To find the hreadth. The beams of a warehouse 
floor are to be of Georgia pine, with a clear bearing between 
the walls of 15 feet, and placed 14 inches from centres, what 
must be the breadth when the depth is 11 inches ? By the 
rule, 3375, the cube of the length, and 1^-, tlie distance from 
centres, and 1*35, the value of y, for Georgia pine, all multi- 
plied together, equals 5315*625 ; and this product divided by 
1331, the cube of the depth, equals 3*994, the required depth, 
or 4 inches. 



FKAMIKG. 241 

Example. — To jmd the depth. The conditions remaining, 
as in last example, what must be the depth when the breadth 
is 3 inches? 5315*625, the said product, divided by 3, the 
breadth, equals 17Y1-875, and the cube root of this, 12*1, or 12 
inches, is the depth required. 

Example. — To jmd the hreadth and depth in a given propor- 
tion^ say, 0-35 to I'O. 5315'625 aforesaid, divided by 0*35, the 
value of r^ equals 15187*5, the square root of w^hich is 121*8, 
and the square root of this square root is 11*04, or 11 inches, 
the required depth. And 11*01: multiplied by 0"35, the value 
of r, equals 3'864:, the required breadth — 3-g- inches. 

340. — It is sometimes desirable, when the breadth and depth 
of the beams are fixed, or when the beams have been sawed 
and are now ready for use, to know the distance from centres 
at which such beams should be placed, in order that the floor 
be sufficiently stiff. In this case, (175,) transposed, and put- 
ting X = — j-y-, there results 

c = l§ (178.) 

Tliis in words, at length, is, as follows : 

I^ule XLIX. — Multiply the cube of the depth by the 
breadth, both in inches, and divide the product by the cube 
of the length, in feet, multiplied by the value of x, for dwell- 
ings, and for ordinary stores, or by y for warehouses ; and the 
quotient will be the distance apart from centres in feet. 

Example. — A span of 17 feet, in a dwelling, is to be covered 
by white pine beams, 3 X 12 inches, at what distance apart 
from centres must they be placed ? By the rule, 1728, the 
cube of the depth, multiplied by 3, the breadth, equals 5184. 
The cube of 17 is 4913, this by 1*0, the value of aj, tor white 
pine, equals 4913. The aforesaid 5184, divided by this, 4913, 
equals 1*055 feet, or 1 foot and | of an inch. 

341. — Where chimneys, flues, stairs, etc., occur to interrupt 

31 



242 



AMERICAN HOIJSE-CAEPENTEE. 




Fig. 



the bearing, the beams are framed into a piece, 5, {Fig. 227,) 
called a header. The beams, a a, into which the header is 
framed, are called trimmers or carriage-lyeams. These framed 
beams require to be made thicker than the common beams. 
The header must be strong enough to sustain one-half of the 
weight that is sustained upon the tail beams, c c, (the wall at 
the opposite end or another header there sustaining the other 
half,) and the trimmers must each sustain one-half of the 
weight sustained by the header in addition to the weight it 
supports as a common beam. It is usual in practice to make 
these framed beams one inch thicker than the common beams 
for dwellings, and two inches thicker for heavy stores. This 
practice in ordinary cases answers very well, but in extreme 
cases these dimensions are not proper. Rules applicable gene- 
rally must be deduced from the conditions of the case — the 
load to be sustained and the strength of the material. 

342. — ^For the header, formula (68,) Table Y., is applicable. 
The weight, represented by w^ is equal to the superficial area 
of the floor supported by the header, multiplied by the load 
on every superficial foot of the floor. This is equal to the 
length of the header multiplied by half the length of the tail 
beams, and by 86 pounds for dwellings and ordinary stores, or 



FEAMING. 243 

by 500 pounds for warehouses. Calling the length of the tai] 
beams, in feet, ^, formula (68,) becomes 

Tlien if/ equals 86, and n equals 0*03, there results 

^ = ^ (m.) 

This in words, is, as follows : 

Rule L. — Multiply 900 times the length of the tail beams 
by the cube of the length of the header, both in feet. The 
product, divided by the cube of the depth, multiplied by the 
value of E^ Table II., will equal the breadth, in inches, for 
dwellings or ordinary stores. 

Examjple. — A header of white pine, for a dwelling, is 10 feet 
long, and sustains tail beams 20 feet long, its depth is 12 
inches, what must be its breadth ? By the rule, 900 X 20 x 
10" = 18,000,000. This, divided by (12' X 1750 =) 3,024,000, 
equals 5*95, say 6 inches, the required breadth. 

For heavy warehouses the rule is the same as the above, 
only using 1550 in the place of the 900. This constant may 
be varied, at discretion, to anything between 900 and 5000, in 
accordance with the use to which the floor is to be put. 

343. — In regard to the trimmer or carriage beam, formula 

(136,) Table YIIL, is applicable. The load tlirown upon the 

trimmer, in addition to its load as a common beam, is equal 

to one-half of the load on the header, and therefore, as has 

been seen in last article, is equal to one-half of the superticial 

area of the floor, supported by the tail beams, multiplied by 

the weight per superficial foot of the load upon the floor; 

therefore, w^hen the length of the header, in feet, is represented 

1 n 
by/, and the length of the tail beam by 7^, w equals — x ^x/*, 

2 2t 

equals \fj n, and therefore (136,) of Table YIIL, becomes 

, fj m n^ 

^'^ 8adU 



244: AMERICAN HOUSE-CAKPENTER. 

t 

equals the additional tliickness to be given to a common 
beam when used as a trimmer, and for dwellings when f 
equals 86 and a equals 0*3, this part of the formula reduces to 
286 §, or, for simplicity, call it 300, which would be the same 
as fixing/ at 90 instead of 86. Then we have 

1 = '-^^^ (180.) 

This, in words, is as follows : 

Hule LI. — 1^07^ dwellings. Multiply 300 times the length 
of the header by the square of the length of the tail beams, 
and by the difference in length of the trimmer and tail beams, 
all in feet. Divide this product by the square of the depth in 
inches, multiplied by the length of the trimmer in feet, and by 
the value of S, Table YL, and the quotient added to the thick- 
ness of a common beam of the floor, will equal the required 
thickness of the trimmer beam. 

Example. — A tier of 3 x 12 inch beams of white pine, hav- 
ing a clear bearing of 20 feet, has a framed well-hole at one 
side, of 5 by 12 feet, the header being 12 feet long, what must 
be the thickness of the trimmer beams ? By the rule, 300 x 
12 X V^^ X 5, divided by the product of 12' x 20 x 390, equals 
3*6, and this added to 3, the thickness of one of the conimon 
beams, equals ^'^.^ the breadth required, 6-|- inches. 

For stores and warehouses the rule is the same as the above, 
only the constant, 300, must be enlarged in proportion to the 
load intended for the floor, making it as high as 1600 for 
heavy warehouses. 

344. — "When a framed opening occurs at any point removed 
from the wall, requiring two headers, then the load from the 
headers rest at two points on the carriage beam, and here for- 
mula (141.) Table YIII., is applicable. In this special case 
this formula reduces to 

^ = swi ^^^^'^ 



FRAMING. 245 

where h equals the additional thickness, in inches, to be given 
to the carriage beam over the thickness of the common beams ; 
y, the length of the header, in feet ; m and h the length, in 
feet, respectively, of the two sets of tail beams, and m-]r n = k 

The constant in the above, (181,) is for dwellings ; if the 
floor is to be loaded more than dwelling floors, then it must be 
increased in proportion to the increase of load up to as high as 
1600 for warehouses. 

Rule LII. — Trimmer heams for framed openings occurring 
so as to require two headers. Multiply the square of the 
length of each tail beam by the difference of length of the tail 
beam and trimmer, all in feet, and add the products ; multiply 
their sum by 300 times the length, in feet, of the header, and 
divide this product by the product of the square of the depth, 
in inches, by the length, in feet, and by the value of /S, Table 
YI. ; and the quotient, added to the thickness of a common 
beam of the tier, will equal the thickness of the trimmer 
beams. 

Example. — A tier of white pine beams, 4 X 14 inches, 20 
feet long, is to have an opening of 5 x 10 feet, framed so that 
the length of one series of tail beams is 7 feet, the other 8 feet, 
what must be the breadth of the trimmers ? Here, {T X 13) 
+ (8^^ X 12) equals 1405. This by 300 x 10 equals 4,215,000. 
This divided by 1,528,800 {= 14' + 20 X 390) equals 2-75, and 
this added to 4, the breadth, equals 6*75, or 6f, the breadth 
required, in inches. 

345. — Additional stiffness is given to a floor by the insertion 
of bridging strips, or struts, as at a a, {Fig. 228.) These pre- 
vent the turning or twisting of the beams, and when a weight 
is placed upon the floor, concentrated over one beam, they 
prevent this beam from descending below the adjoining beams 
to the injury of the plastering upon the underside. It is usual 
to insert a course of bridging at every 5 to 8 feet of the length 



246 



AMERICAN HOUSE-CARPENTEK. 




Fig. 228. 



of the beam. Strips of board or plank nailed to the underside 
of the floor beams to receive the lathing, are termed cross- 
furring^ and should not be over 2 inches wide, and placed 12 
inches from centres. It is desirable that all furring be narrow, 
in order that the clinch of the mortar be interrupted but little. 
When it is desirable to prevent the passage of sound, the 
openings between the beams, at about 3 inches from the upper 
edge, are closed bj short pieces of boards, which rest on elects, 
nailed to the beam along its whole length. This forms a floor, 
on which mortar is laid from 1 to 2 inches deep. This is 
called deafening. 

346. — When the distance between the walls of a building is 
great, it becomes requisite to introduce girders, as an addi- 
tional support, beneath the beams. The dimensions of girders 
may be ascertained by the general rules for stiffness. For- 
mulas (Y2,) (73,) and (74,) Table Y., are applicable, taking /*, 
at 86, for dwellings and ordinary stores, and increased in pro- 
portion to the load, up to 500, for heavy warehouses. When 
but one girder occurs, in the length of the beam, the distance 
from centres, <?, is equal to one-half the length of the beam. 

347. — ^When the breadth of a girder is more than about 12 
inches, it is recommended to divide it by sawing from end to 
end, vertically through the middle, and then to bolt it to 



FEAMTN-G. 24:7 

gether with the sawn sides outwards. This is not to strength- 
en the girder, as some have supposed, but to reduce the size 
of the timber, in order that it may dry sooner. The opera- 
tion affords also an opportunity to examine tlie heart of the 
stick — a necessary precaution; as large trees are frequently 
in a state of decay at the heart, although outwardly they are 
seemingly sound. When the halves are bolted together, thin 
slips of wood should be inserted between them at the several 
points at which they are bolted, in order to leave sufficient 
space for the air to circulate between. This tends to prevent 
decay ; v/hich will be found first at such parts as are not 
exactly tight, nor yet far enough apart to permit the escape 
of moisture. 

3iS. — When girders are required for a long bearing, it is 
usual to truss them ; that is, to insert between the halves two 
pieces of oak which are inclined towards each other, and 
which meet at the centre of the length of the girder, like the 
rafters of a roof-truss, though nearly if not quite concealed 
within the girder. This, and many similar methods, though 
extensively practised, are generally worse than useless ; since 
it has been ascertained that, in nearly all such cases, the ope- 
ration has positively weakened the girder. 

A girder may be strengthened by mechanical contrivance, 
when its depth is required to be greater than any one piece of 




Fig. 229. 



2i8 AMEBIC A2?" HOUSE- CARrEXTER. 

timber will allow. Fig. 229 shows a very simple yet mvalu 
able method of doing this. The two pieces of which the gir- 
der is composed are bolted, or pinned together, having keys 
inserted between to prevent the pieces from sliding. The 
keys should be of hard wood, well seasoned. The two pieces 
should be about equal in depth, in order that the joint be- 
tween them may be in the neutral line. (See Art. 317.) The 
thickness of the keys should be about half their breadth, and 
the amount of their united thicknesses should be equal to a 
trifle over the depth and one-third of the depth of the girder. 
Instead of bolts or pins, iron hoops are sometimes used ; and 
when they can be procured, they are far preferable. In this 
case, the girder is diminished at the ends, and the hoops 
driven from each end towards the middle. 

34:9. — Beams may be spliced, if none of a sufficient length 
can be obtained, though not at or near the middle, if it can 
be avoided. (See Art. 281.) Girders should rest from 9 to 
12 inches on the wall, and a space should be left for the air 
to circulate around the ends, that the dampness may evapo- 
rate. Floor-timbers are supported at their ends by walls of 
considerable height. They should not be permitted to rest 
upon intervening partitions, which are not Hkely to settle as 
much as the walls ; otherwise the unequal settlements will 
derano^e the level of the floor. As all floors, however well- 
constructed, settle in some degree, it is advisable to frame 
the beams a little higher at the middle of the room than at 
its sides, — as also the ceiling-joists and cross-furring, when 
either are used. In single floors, for the same reason, the 
rounded edge of the stick, if it have one, should be placed 
uppermost. 

If the floor-plank are laid down temporarily at first, and left 
to season a few months before they are finally driven together 
and secured, the joints will remain much closer. But if the 
edo-es of the plank are planed after the first laying, they will 



FRAMING. 249 

shrink again ; as it is the nature of wood to shrink after every 
planing however dry it may have been before. 



PARTITIONS. 

o50. — Too httle attention has been given to the construction 
of this part of the frame-work of a house. The settling of 
floors and the cracking of ceilings and walls, which disfigure 
to so great an extent the apartments of even our most costly 
houses, may be attributed almost solely to this negligence, A 
square of partitioning weighs nearly a ton, a greater weight, 
when added to its customary load, such as furniture, storage, 
&c., than any ordinary floor is calculated to sustain. Hence 
the timbers bend, the ceilings and cornices crack, and the 
wholo interior part of the house settles ; showing the necessity 
for providing adequate supports independent of the floor- 
timbers. A partition should, if practicable, be supported by 
the walls with which it is connected, in order, if the walls set- 
tle, that it may settle with them. This would prevent the 
separation of the plastering at the angles of rooms. For the 
same reason, a firm connection with the ceiling is an im- 
portant object in the construction of a partition. 

351. — The joists in a partition should be so placed as to dis- 
charge the weight upon the points of support. All oblique 
pieces in a partition, that tend not to this object, are much 
better omitted. Fig. 230 represents a partition having a door 
in the middle. Its construction is simple but effective. Fig. 
231 shows the manner of constructing a partition having doors 
near the ends. The truss is formed above the door-heads, 
and the lower parts are suspended from it. The posts, a and 
5, are halved, and nailed to the tie, c d^ and the sill, e f. The 
braces in a trussed partition should be placed so as to form, as 
near as possible, an angle of 40 degrees with the horizon. In 
partitions that are intended to support only their own weight, 

32 



250 



AMERICAN HOUSE-CARPENTEE. 



n n 

1 


1 

1 

A ' v\ ■ . . . , 


IJw^i . ,-1- iK. 


U lJ 



Fig. 280. 




the principal timbers may be 3 X 4 inches for a 20 feet span, 
3i X 5 for 30 feet, and 4x6 for 40. The thickness of the 
filling-in stuff may be regulated according to what is said at 
Art. 345, in regard to the width of furring for plastering. 
The filling-in pieces should be stiffened at about every three 
feet by short struts between. 

All superfluous timber, besides being an unnecessary load 
upon the points of support, tends to injure the stability of the 
plastering ; for, as the strength of the plastering depends, in a 
great measui^e, upon its clinch, formed by pressing the mortar 



FRAMING. 



251 



througli the space between the laths, the narrower the surface, 
therefore, upon which the laths are nailed, the less will be the 
quantity of plastering unclinched, and hence its greater secu- 
rity from fractures. For this reason, the principal timbers of 
the partition should have their edges reduced, by chamfering 
off the corners. 



^. 



^ 



^ 



^s 



^h 



Fig. 282. 

S52. — When the principal timbers of a partition require to 
be large for the purpose of greater strength, it is a good plan 
to omit the upright filling-in pieces, and in their stead, to 
place a few horizontal pieces ; in order, upon these and the 
principal timbers, to nail upright battens at the proper dis- 
tances for lathing, as in Fig. 232. A partition thus con- 
structed requires a little more space than others ; but it has 
the advantage of insuring greater stability to the plastering, 
and also of preventing to a good degree the conversation of 
one room from being heard in the other. When a partition is 
required to support, in addition to its own weight, that of a 
floor or some other burden resting upon it, the dimensions of 
the timbers may be ascertained, by applying the principles 
which regulate the laws of pressure and those of the resistance 
of timber, as explained at the first part of this section. The 
following data, however, may assist in calculating tlie amount 
of pressure upon partitions : 



352 AMERICAN HOUSE-CAEPENTEE. 

Wli.te pine timber weighs from 22 to 32 pounds per cubic 
foot, varying in accordance with tlie amount of seasoning it 
has had. Assuming .it to weigh 30 pounds, the weight of the 
beams and floor plank in every superficial foot of the flooring 
will be, when the beams are 

3x8 inclies, and placed 20 inches from centres, 6 pounds. 
3 X 10 " " " 18 " " " 1i *' 

3 X 12 " " " 16 " " " 9 

3 X 12 " " " 12 " " " 11 " 

4 X 12 " " " 12 " " " 13 
4 X 14 " " " 14 " " " 13 

In addition to the beams and plank, there is generally the 
jpldstering of the ceiling of the apartments beneath, and some- 
times the deafening. Plastering may be assumed to weigh 9 
pounds per superficial foot, and deafening 11 pounds. 

Hemlock weighs about the same as white pine. A parti- 
tion of 3 X 4 joists of hemlock, set 12 inches from centres, 
therefore, will weigh about 2|- pounds per foot superficial, and 
when plastered on both sides, 20-|- pounds. 

353. — When floor beams are supported at the extremities, 
and by a partition or girder at any point between the extre- 
mities, one-half of the weight of the whole floor will then be 
supported by the partition or girder. As the settling of parti- 
tions and floors, which is so disastrous to plastering, is fre- 
quently owing to the shrinking of the timber and to ill-made 
joints, it is very important that the timber be seasoned and 
the work well executed. Where practicable, the joists of a 
partition ought to extend down between the floor beams to 
the plate of the partition beneath, to avoid the settlement con- 
sequent upon the shrinkage of the floor beams. 

ROOFS.* 

354. — In ancient biildings, the Norman and the Gothic, the 

• See also Art. 238. 



FKAMTN-G. 



253 



walls and buttresses were erected so massive and firm, that it 
was customary to construct their roofs without a tie-beam; 
the walls being abundantly capable of resisting the lateral 
pressure exerted by the rafters. But in modern buildings, the 
walls are so slightly built as to be incapable of resisting 
scarcely any oblique pressure ; and hence the necessity of 
constructing the roof so that all oblique and lateral strains 
may be removed ; as, also, that instead of having a tendency 
to separate the walls, the roof may contribute to bind and 
steady them. 

355. — In estimating the pressures upon any certain roof, for 
the purpose of ascertaining the proper sizes for the timbers, 
calculation must be made for the pressure exerted by the wind, 
and, if in a cold climate, for the weight of snow, in addition to 
the weight of the materials of which the roof is composed. 
The weight of snow will be of course according to the depth 
it acquires. Snow weighs 8 lbs. per cubic foot, and more 
when saturated with water.- In a severe climate, roofs ought 
to be constructed steeper than in a milder one, in order that 
the snow may have a tendency to slide off before it becomes 
of sufiicient weight to endanger the safety of the roof. The 
inclination should be regulated in accordance with the qualities 
of the material with which the roof is to be covered. The 
following table may be useful in determining the smallest in- 
clination, and in estimating the weight of the various kinds 
of covering : 



Material. 


Inclination. 


Weight upon a 
square foot. 


Tin 

Sr. : : : : 

Zinc 

Short pine shingles 
Long cypress shingles . 
Slate 






Rise 1 inch to a foot 
" 1 " " " 
" 2 inches " " 
" 3 " " " 
" 5 .« u << 
a 6 " " " 
« 6 " " " 


f to H lbs. 

1 to li " 

4 to 7 " 
li to 2 " 
H to 2 '• 

2 to 8 « 

5 to 9 " 



2d4: AMERICAN HO :SE-CAEPENTEE. 

The weight of the covering, as above estimated, is that of 
the material only, with the weight of whatever is used to fix 
it to the roof, siich as nails, &c. What the material is laid 
on, such as plank, boards or lath, is not included. The weight 
of plank is about 3 pounds per foot superficial ; of boards, 2 
pounds; and lath, about a half pound. 

356. — The weights and pressures on a roof arise from the 
roofing, the truss, the ceiling, wind and snow, and may be 
stated as follows : 

I^irst, the Roofing. — On each foot superficial of the inclined 
surface. 

Slating will weigh about 7 lbs. 

Roof plank, 1:^ inches thick . . « « " 2''7 " 

Roof beams or jack rafters . . « « « 2-3" 

Total, 12 lbs. 

'This is the weight per foot on the inclined surface ; but it is 
desirable to know how much per foot, measured horizontally^ 
this is equal to. The horizontal measure of one foot of the 
inclined surface is equal to the cosine of the angle of inclina- 
tion. Therefore, 

cos. \ \ \\ p \ w — -^- ; 
cos. 

where jp represents the pressure on a foot of the inclined sur- 
face, and w the weight of the roof per foot, measured horizon- 
tally. The cosine of an angle is equal to the base of the right- 
angled triangle divided by the hypothenuse, which in this 
case would be half the span divided by the length of the 

rafter, or — , where s is the span, and I the length "f the 

rafter. Hence, 

p _ 2^ _^lp 

COS. ^ s ' 

or, twice the pressure per foot of inclined surface, multiplied 



FRAiVIEN-G. 25 c- 

by the length of the rafter, and diviaed by the span, will give 
the weight per foot measured horizontally ; or, 

24:- =w (182.) 

s 

equals the weight per foot, measured horizontally, of the roof 

beams, plank, and covering for a slate roof. 

/Seoo?id, the Truss. — The weight of the framed truss is nearly 

in proportion to the length of the truss, and to the distance 

apart at which the trusses are placed. 

w = 5-2 cs (183.) 

equals tlie weight, in pounds, of a white pine truss with iron 

suspension rods and a horizontal tie beam, near enough for 

the requirements of our present purpose ; where s equals the 

length or span of the truss, and c the distance apart at which 

the trusses are placed, both in feet. It is desirable to know 

how much this is equal to per foot of the area over which the 

truss is to sustain a covering. This is found by dividing the 

weight of the truss by the span, and by the distance anart 

from centres at which the trusses are placed ; or, 

^^^ = 5-2 =. 10 (184.) 

cs ' 

equals the weight in pounds per foot to be allowed for the 
truss. 

Thirds the Ceiling. — The weight supported by the tie beams, 
viz. : that of the ceiling beams, furring and plastering, is about 
9 pounds per superficial foot. 

Fourth^ the Wind. — ^The force of wind has been known as 
high as 50 pounds per superficial foot against a vertical sur- 
face. The effect of a horizontal force on an inclined surface 
is in proportion to the sine of the angle of inclination, the ef- 
fect produced being in the direction at right angles to the in- 
clined surface. The force thus acting may be resolved into 
forces acting in two directions — the one horizontal, the other 
vertical ; the former tending, in the case of a roof, to thrust 



256 AMEEICAIs HOrSE-CAKPEXTEE. 

aside the walls on which the roof rests, and the latter^ acting 
directly on the materials of which the roof is constructed — 
this latter force being in proportion to the sine of the angle 
of inclination multiplied by the cosine. This is the vertical 
effect of the wind upon a roof, without regard to the surface 
it acts upon. The wind, acting horizontally through one foot 
superficial of vertical section, acts on an area of inclined sui-- 
face equal to the reciprocal of the sine of inclination, and the 
horizontal measurement of this inclined surface is equal to the 
cosine of the angle of inclination divided by the sine. This is 
the horizontal measurement of the inclined surface, and the 
vertical force acting on this surface is, as above stated, in pro- 
portion to the sine multiplied by the cosine. Combining these, 
it is found that the vertical power of the wind is in proportion 
to the square of the sine of the angle of inclination. There- 
fore, if the power of wind against a vertical surface be taken 
at 50 pounds per su2:)erficial foot, then the vertical effect on a 
roof is equal to 

wj = 50 sin.^ = 50 ^ (185.) 

for each piece of the inclined surface, the horizontal measure- 
ment of which equals one foot ; where I equals the length of 
the rafter, and h the height of the roof 

Fifths Snow.—T\\Q weight of snow will be in proportion to 
the depth it acquires, and this will be in proportion to the 
rigour of the climate of the place at which the building is to 
be erected. Upon roofs of most of the usual inclinations, 
snow, if deposited in the absence of wind, will not slide off. 
When it has acquired some depth, and not till then, it will 
have a tendency, in proportion to the angle of inclination, to 
slide off in a body. The weight of snow may be taken, there- 
fore, at its weight per cubic foot, 8 pounds, multiplied by the 
depth it is usual for it to acquire. This, in the latitude of !N'ew 
York, may be stated at about 2^ feet. Its weight would, 



FRAMING. 257 

therefore, be 20 pounds per foot superficial, measured horizon- 
tally. 

357. — ^There is one other cause of strain upon a roof; namely, 
the load that may be deposited in the roof when used as a room 
for storage, or for dormitories. But this seldom occurs. When 
a case of this kind does occur, allowance is to be made for it 
as shown in the article on floors. But in the general rule, now 
under consideration, it may be omitted. 

358. — ^The following, therefore, comprehends all the pressures 
or weights that occur on roofs generally, per foot superficial ; 

I 

For roof beams, plank, and slate (182) . . . . 24 - lbs. 

" the truss (184) . 5*2 « 

" ceiling 9 " 

" wind (185) ^0 7a " 

•• snow, latitude of New York 20 " 

Having found the weight per foot, the total weight for any 
part of the roof is found by multiplying the weight per foot by 
the area of that part. This process will give the weight sup- 
ported by braces and suspension rods, and also that supported 
by the rafters and tie beam. But in these last two, only half 
of the pressure of the vnnd is to be taken, for the wind will 
act only on one side of the roof at the same time. 

The vertical pressure on the head of a brace, then, equals 

TF = 4 c n(6 - + 8-55 + 12*5 ^) (186.) 

f I h^ 

And W =: cpn^ where p equals 4^6- + 8*55 + 12*5 — r, 

equals the weight per foot. 

And the aggregate load of the roof on each truss equals 

TT = 4 c 5 (6 ^ + 8-55 + 6-25 ^) (187.) 

A nd W=cqs^ where ^ = 4^6 - + 8-55 H- 6*25 j\ equals the 

33 



258 AMERICAN HOUSE-CABPENTEK. 

weight per foot ; where c equals the distance apart from centres 
at which the trusses are placed ; n the distance horizontally 
between the heads of the braces, or, if these are not located at 
equal distances, then n is the distance horizontally from a point 
half-way to the next brace on one side to a point half-way to 
the next brace on the other side ; I the length of the rafter ; % 
the span, and A the height — all in feet. 

359. — By the parallelogram of forces, the weight of the roof 
is in proportion to the oblique thrust or pressure in the axis of 
the rafter, as twice the height of the roof is to the length of the 
rafter; or, 

TT : 7? :: 2A : Z, or 

2A : I \\W \ R=:-^, (188.) 

where R equals the pressure in the axis of the rafter. And 
the weight of the roof is in proportion to the horizontal thrust 
in the tie beam, as twice the height of the roof is to half the 
span; or, 

W \ ^:: 2A • -^5<=>i' 

2A : I :: TT : H= E|, (189.) 

where H equals the horizontal thrust in the tie beam ; the 
value of W in (188) and (189) being shown at (187), and (187) 
being compounded as explained in Art. 356. The weight 
is that for a slate roof If other material is used for covering, 
or should there be other conditions modifying the weight in 
any particular case, an examination oi Art. 356 will show how 
to modify the formula accordingly. 

360. — The pressures may be obtained geometrically, as 
shown in Fig, 233, where A B represents the axis of the tie 
beam, A C the axis of the rafter, D E and F B the axes of the 
braces, and D G^ FE^ and C B^ the axes of the suspension rods. 
In tliis design for a truss, the distance ^ ^ is divided into three 



FRAMING. 259 

equal parts, and the rods located at tlie two points of division, 
G and E. By this arrangement the rafter AG\s> supported at 
equi-distant points, D and F, The point D supports the rafter 
for a distance extending half-way to A and half-way to F^ and 
the point F sustains half-way to B and half-way to C. Also, 
the point C sustains half-way to F and, on the other rafter, 
half-way to the corresponding point to F, And because these 
jDoints of support are located at equal distances apart, there- 
fore the load on each is the same in amount. On D G make 
D a equal to lOU of any decimally divided scale, and let D a 
represent the load on D^ and draw the parallelogram ah D c. 
Then, by the same scale, D h represents {Art. 258) the pressure 
in the axis of the rafter by the load at D ; also, D c the pressure 
in the brace D E. Draw c d horizontal ; then D d is the ver- 
tical pressure exerted by the brace DF at F, The point F 
sustains, besides the common load represented by 100 of the 
scale, also the vertical pressure exerted by the brace D F * 
therefore, since D a represents the common load on D, F^ or 
6', make Fe equal to the sum o^ D a and D d^ and draw the 
parallelogram Fgef. . Then Fg^ measured by the scale, is 
the pressure in the axis of the rafter caused by the load at F, 
and Ff is the load in the axis of the brace F B, Draw fh 
horizontal ; then Fh is the vertical pressure exerted by the 
brace F B Sit B. The point G, besides the common load re- 
presented by D <2, sustains the vertical pressure Fh caused by 
the brace FB, and a like amount from the corresponding brace 
on the opposite side. Therefore, make Gj equal to the sum of 
D a and twice Fh, and draw jk parallel to the opposite rafter. 
Then Gkis the pressure in the axis of the rafter at G. This is 
not the only pressure in the rafter, although it is the total 
pressure at its head G. At the point F, besides the pressure 
Gk, there is Fg, At the point Z>, besides these two pressures, 
there is the pressure D h. At the foot, at A, there is still an 
additional pressure : while the point D sustains the load half- 



260 AMERICAN HOrSE-CAEPEXTER. 

way to F and half-way to A^ the point A sustains the load 
half-way to D. This load is, in this case, just half the load at 
D. Therefore draw A m vertical, and equal to 50 of the scale, 
or half of D a. Extend C A to I ; draw m I horizontal. Then 
Al h the pressure in the rafter at A caused by the weight of 
the roof from A half-way to D. Now the total of the pressures 
in the rafter is equal to the sum of Al -{- Dh -f- Fg added to 
C k. Therefore make Is n equal to the sum oi Al -\- Dl -\- Fg^ 
and draw n o parallel with the opposite rafter, and nj hori- 
zontal. Then Co, measured by the same scale, will be found 
equal to the total weight of the roof on both sides of G B. If 
D a = 100 represent the portion of the weight borne by the 
point D, then Co, representing the whole weight of the roof, 
should equal 600, (as it does by the scale,) for D supports just 
one-sixth of the whole load. As Cn is the total oblique thrust 
in the axis of the rafter at its foot, therefore nj is the horizon- 
tal thrust in the tie beam. 

361. — In stating the amount of pressures in the above as 
being equal to certain lines, it was so stated with the under- 
standing that the lines were simply in proportion to the weights. 
To obtain the weight represented by a line, multiply its length 
(measured by the scale used) by the load resting at D, (or at 
i^or C, as these are all equal in this example,) and divide the 
product by 100, and the quotient will be the weight required. 
For, as 100 of the scale is to the load it represents, so is any 
other dimension on the same scale to the load it represents. 

Zm.— Example. Let A B {Fig. 233) equal 26 feet, CB 18 
feet, and J. (7 29 feet, and AG, GF, and FB, each 8f feet. 
Let the trusses be placed 10 feet apart. Then the weight on 
Z>, for the use of the braces and rods, is, per (186), equal to 

4^71(^6^ f 8-55+ 12i|-'j 
= 4 X 10 X 81(6 X g + 8-55 + 12J x g-,) 



FRAMING. 2(3/ 

= Sttet X 14-398 
= 4991-3. 
This is the common load at the points D^ F^ and (7, and eacli 
of the lines denoting pressures multiplied by it and divided by 

4991*3 
100, or multiplied by the quotient of -zr— — 49 "91 3. the pro- 
duct will be the weight required. 49"913 may be called 50, 
for simplicity ; therefore the pressure in the brace D E equals 
112 X 50 = 5600 pounds, and in the brace F B, 140 X 50 :- 
7000 pounds, and in like manner for any other strain. For 
the rafters and tie beam the total weight, as per (187), equals 

4^^(6^+8-55 -1-61 1) 



= 4 X 10 X 52(6 X ^ + 8-55 + 6i X ^|,) 

= 2080 X 13-148 
= 27343-68 pounds. 
This is the total weight of the roof supported by one truss. 
The oblique thrust in the rafter A G is, per (188), equal to 

nF_29 X 27343-68 
2 A 2x13 

= 30498-72 pounds. 

To obtain this oblique thrust geometrically : Co {Fig. 233) 
represents the weight of the roof, and measures 600 by the scale ; 
and the line Gn^ representing the oblique thrust, measures 670. 
By the proportion, 600 : 670:: 27343-68 : 30533-8, = the 
oblique thrust. The result here found is a few pounds more 
than the other. This is owing to the fact that tlie line Gn'i^ 
not exactly 670, nor is the length of the rafter precisely 29 feet. 
Were the exact dimensions used in each case the results would 
be identical ; but the result in either case is near enough for 
the purpose. 

The horizontal strain is, per (189), equal to 



262 ' AMERICAN HOUSE-CAEPENTEK. 

Ws __ 27343-68 X 52 
4A ■" 4x 13 

= 27343-68 pounds. 

The result gives the horizontal thrust precisely equal to the 
weight. This is as it should be in all cases where the height 
of the roof is equal to one-fourth of the span, but not other- 
wise ; for the result depends (189) upon this relation of the 
height to the span. Geometrically, the result is the same, for 
Co and nj (Fig. 233,) representing the weight and horizontal 
thrust, are precisely equal by measurement. 

363. — ^The weight at the head of a brace is sustained partly 
at the foot of the brace and partly at the foot of the rafter. 
The sum of the vertical effects at these two points is just equal 
to the weight at the head of the brace. The portion of the 
weight sustained at either point is in proportion, inversely, to 
the horizontal distance of that point from the weight ; there- 
fore, 

V=W^, (190.) 

where Y equals the vertical effect at the foot of the brace ; TF, 

the weight at the head of the brace ; g^ the horizontal distance 

from the foot of the rafter to the head of the brace ; and a^ the 

distance from the same point to Xh^foot of the brace. 

364. — For the oblique thrust in the brace : from the triangle 

Ffh {Fig, 233,) 

Fh : Ff:: sin. : rad. 

sin. : rad. :: V : T; 
tlierefore, 

T^X-=^vi (191.) 

sm. h 

where 2^ equals the oblique thrust in the brace; F", the verti- 
cal pressure caused by T at the foot of the brace (190) ; and 
I and h the length and height respectively of the brace. 

d66.—Fxample. Brace D F, Fig. 233. In this case, W 
equals the product of the weight per superficial foot, multi- 



FRAMING. 263 

plied by the area supported at the point i>, equals 5000 
pounds, {Art. 362.) The length ^ equals 8f feet, and a equals 
ITi feet. Therefore (190), 

F=Tr^=5000 X 1^=2500 pounds 

equals ihe vertical pressure at £! caused by the brace D£J. 
Then, for the oblique thrust, I equals 9-6 feet, and h equals 4*3 
feet. Therefore, from (191), 

T=vl=: 2500 X ^ =- 55814 pounds 
h 4*3 

equals the oblique thrust in the brace D E. In Art. 362 it 

was found to be 5600. The discrepancy is owing to like causes 

of want of accuracy in the case of the rafter, as explained in 

Art. 362. 

Another Example. — Brace EB^ Eig. 233. In this case, W 

equals the product of the weight per superficial foot, multiplied 

by the area supported by the point E^ added to the vertical 

strain caused by the brace D E. From Art. 362 the weight 

of roof on E equals 5000 pounds, and the vertical strain from 

brace D E is, as just ascertained, =: 2500, total 7500, equals 

W. The length, ^, equals two-thirds of 26 feet, equals 17J, 

and a equals 26 feet. Therefore, from (190), 

V=:W^ = 7500 x-^ = [.000 
a 26 

equals the vertical effect at E caused by the brace EE. 

Then, for the oblique thrust in the brace, I equals 12*2, and h 

equals 8f . Therefore, from (191) 

7 19'9 

T=V~ = 5000 x~= 7038-5 
A of 

equals the oblique thrust or strain in tlie axis of the brace. It 
was 7000 by the geometrical process, {Art. 362.) 

366. — ^The strain upon the first rod, E G, equals simply the 
weight of the ceiling supported by it, adiled to the part of the 
tie beam it sustains. The weight of the tie beam will equal 



264 



A^IERICAN HOUSE-CAEPENTER. 




F\e. 233. 



FRAMING. 265 

about one pound per superficial foot of the ceiling. The weight 
per foot for the ceiling is stated (see Art. 356 third, and 358,) 
at 9 pounds. To this add 1 pound for the tie beam, and the 
sum is 10. Then ' 

]Sr= 10 en. (192.) 

The strain on the second tie rod equals the weight of ceiling 
supporte-d, = iT, added to the vertical effect of the strain in the 
brace it sustains, [see (190)] or equal to 

= 10 on + V. (193.) 

The strain on the third rod is equal to iT, added to the ver- 
tical effect of the strain in the brace it sustains, and this is the 
strain on any rod. The first rod has no brace to sustain, and 
the middle rod sustains two braces. In this case the strain 

equals 

77 ^10 en + 2V. (194.) 

It may be observed that V represents the vertical strain 
caused by that brace that is sustained, by the rod under consi- 
deration ; and, as the vertical strain caused by any one brace is 
more than that caused by any other brace nearer the foot of 
the rafter, therefore the V of (193) is not equal to the V of 
(194). Hence a necessity for care lest the two be confounded 
and thus cause error. 

B67.— Samples. The rod D G (Fig. 233) has a strain 
which equals (192) 

]^= 10 on = 10 X 10 X S% = 867 pounds. 

The strain on rod i^^ equals (193) 

O = 10Gn-\-V=S67 -\- 2500 = 3367 pounds. 

The strain on rod GB, the middle rod, equals (194) 

17=10 on -{- 2V= S67 -{- 2 x 5000 = 10867 pounds. 
368. — The load, and the strains caused thereby, having 
been discussed, it remains to speak of the resistance of the ma- 
terials. 

Fii'st, of the Rafter. — Generally this piece of timber is so 
pinioned by the roof beams or purlins as to prevent any late- 

34 



266 AXEEICA2S H0U5E-CAEPEXTZE. 

ral moTement, and the braces keep it from deflection; there- 
fore it is not liable to yield by flexure. Hence the manner of 
its yielding, when overloaded, will be by crushing at the ends, 
or it will crush the tie beam against which it presses. The 
fibres of timber yield much more readily when pressed toge- 
ther by a force acting at inght angUs to the direction of their 
length, than when it acts in a line with their length. 

The value of timber subjected to pressure in these two ways 
is shown in Art. 292, Table I., the value per square inch of 
the first stated resistance being expressed by P, and that of 
the other by C. Timber pressed in an oblique direction yields 
with a force exceeding that expressed by P, and less than that 
by C. When the angle of inclination at which the force acts 
is just 45 "^j then the force will be an average between P and 
C. And for any angle of inclination, the force will vary in- 
versely as the angle ; approaching P as the angle is enlarged, 
and approaching C as the angle is diminished. It will be 
equal to C when the angle becomes zero, and equal P when 
the angle becomes 90°. The resistance of timber per square 
inch to an oblique force is therefore expressed by 

M = P + ^{C-P), (195.) 

where A° equals the complement of the angle of inclination. 
In a roof, A° is the acute angle formed by the rafter with 
a vertical line. If no convenient instrument be at hand to 
measure the angle, describe an arc upon the plan of the 
truss — thus: with CB {Fig. 233) for radius, describe the 
arc B g^ and get the length of this arc by stepping it off with 
a pair of divider. Then 

where a equals the length of the arc, and h equals B (7, the 
height of the roof. Therefore, 



FRAMING. 267 

Jf=P + 0-63||((7-iO (196.) 

equals the value of timber per square inch in a tie beam, C 
and P being obtained from Table L, Art 292. When G for 
the kind of wood in the tie beam exceeds G set opposite the 
kind of wood in the rafter, then the latter is to be used in the 
rules instead of the former. 

369. — Having obtained the strain to which the material is 
subjected in a roof, and the capability of the material to resist 
that strain, it only remains now to state the rules for determin- 
ing the dimensions of the material. 

370. — ^To obtain the dimensions of the rafter : — It has been 
shown that the strain in the axis of the rafter equals (188), 

R = wX,. 
2A 

This is the strain in pounds. Timber is capable of resisting 

effectually, in every square inch of the surface pressed (196), 

P + 0-63f ^(G-P) pounds. 

And when the strain and resistance are equal, 

^ = 56^ [P 4- 0-63||((7 - P)], 

Vi^here h and d are respectively the breadth and depth of the 
rafter. Hence 

I d = ^ . (197.) 

P + 0-631 H<^-^) 

Example. — {Fig. 233.) The strain in the axis of the rafter 

m this example, ascertained in Art. 362, is 30498*72 pounds. 

If the timber used be white pine, then P = 300 and G= 1200, 

The length of the arc -^^ is 14J feet, and h = 13. Therefore 

M= ■ ^Q^^^-^L = 32-8. 

300 + (0-631 X '-IT X 900) 

This is the area of the abutting surface at the tie beam — 
say 6 by 5^ inches. At least half this amount should be added 



268 AilEKICAN HOrSE-CAKPENTEPw 

to allow for the shoulder, and for cutting at the joints foi 
braces, &c. The rafter may therefore be 6 by 9 inches. 

The above method is based upon the supposition that the 
rafter is effectually secured from flexure by the braces and 
roof beams. Should this not be the case, then the dimensions 
of the rafter are to be obtained by rules in Art. 298, for posts. 
Nevertheless, the abutting surface in the joint is to be deter- 
mined by the above formula (197). 

371. — To obtain the dimensions of the braces : — Usually, 
braces are so slender as to require their dimensions to be ob- 
tained by rules in A7t. 298 ; the strain in the axis of the brace 
having been obtained by formula (191), or geometrically as in 
Art 360. 

The abutting surface of the joint of the brace is to be ob- 
tained, as in the case of the rafter, by formula (195) ; A° be- 
ing the number of degrees contained in the acute angle formed 
by the brace and a vertical line, for the joint at the tie beam ; 
but for the joint at the rafter, A° is the number of degrees 
contained in the acute angle formed by the brace and a line 
perpendicular to the rafter, or it is 90, diminished by the num- 
ber of degrees contained in the acute angle formed by the 
rafter and brace. 

Example, — I^ig. 233, Brace Z>^, of white pine. In this 
brace the strain was found {Art. 362) to be 5600 pounds, the 
length of the brace is 9*6 feet. By Art. 298, the brace is 
therefore required to be 4*18 x 6 inches. For the abutting 
surface at the joints, for white pine, P equals 300 and, 6^1200. 
The angle D ^i^ equals 63^ 26'. By (197) and (195), 

T 5600 



hd 



jP + ^iC-JR) 300 H- [-^' X (1200 - 300)] 

5600 ^ . , 
= 6 inches. 



934-5 
This is the area of the abuttirg surface of the joint at the tie 



FRAMING. 269 

beam. To obtain the joint at the rafter, the angle FB E 
equals 53° 8', and hence 

T 6600 

p + f(^irp) ~ 3^[H«^zf^ X (1200 - 300)] 

= 1?^ ^ 8-375 inches. 

300 + e-^ X 900) 

This is the area of the abutting surface of the joint at the 
rafter. 

Another Example, — Brace FB^ Fig. 233, of white pine, 
12-2 feet long. The strain in its axis is {Art. 362) YOOO 
pounds. By Art. 298, the brace is required to be 5J x 6 
inches. For the abutting surface of the joints, P equals 300, 
G equals 1200, and the angle F B G equals 45° ; therefore, 

l)d = t: — = 9J inches. 

300 + [i X (1200 -300)] 

This is the area of the abutting . surface at the tie beam. For 
the surface at the rafter, the angle G FB equals 71°, and 
90 — 71 = 19, equals the angle to be used in the formula ; 

therefore, 

7000 

300 + [^X (1200-300)] 

This is the area of the abutting surface of the joint at the 
rafter. 

372. — To obtain the dimensions of the tie beam : — A tie 
beam must be of such dimensions as will enable it to resist 
effectually the tensile strain caused by the horizontal thrust of 
the rafter and the cross strains arising from the weight of the 
ceiling, and from any load that may be placed upon it in the 
roof. From (17), Art. 310, 



Id = ^^g 7:t:t7 ^^7^^~ -^^'^ inches, nearly. 



A- ™-^ 
where H equals the horizontal thrust, and from (189), 



270 AXEEICAX HOUSE-CARPEXTEE. 



therefore, 



77 ^^. 



A- — ~ ^^ 



where TT equals the weight of the roof in pounds, as shown 
at (187); 5, the span ; A, the height, both in feet; and T^ a 
constant set opposite the kind of material, in Table III. ; and 
A equals the area of uncut fibres in the tie beam. About 
one-half of this should be added to allow for the requisite cut- 
ting at the joints ; or, the area of the cross section of the tie 
beam should be equal to at least | of the area of uncut fibres ; 
or, when l d equals the area of the tie beam, then 

5^ = 1^. (198). 

Examjple. — ^The weight on the truss at Fig. 233 is shown to 
be {Art 362) 27343*68 pounds, say 27500 pounds ; the span is 
52 feet, the height 13, and the value of T for white pine is 
(Table III.) 2367, therefore 

,7 o TF5 o 27500 X 52 -.^ , . -, 
5J=.t-^ = |x— ^^^3^ = lr4mches 

equals the area of cross section of the tie beam requisite to 
resist the tensile strain. This is smaller, as will be shown, than 
what is required to resist the cross strains, and this will be 
found to be the case generally. The weight of the ceiling is 
9 pounds per superficial foot ; the length of the longest unsup- 
ported part of the tie beam is 8f feet ; then, if the deflection 
per lineal foot be allowed at O'Olo inch, the depth of the tie 
beam will be required ((72), Table Y.) to be 6*14: inches. But 
in order efiectually to resist the strains tie beams are subjected 
to at the hands of the workmen, in the process of framing and 
elevating, the area of cross section in inches should be at 
least equal to the length in feet. Were it possible to guard 
against this cause of strain, the size ascertained by the rule, 6 



FRAMING. 271 

by 6*14:, would be sufficient ; but to resist this strain, the size 
should be 6 by 9. 

There is yet one other dimension for the tie beam required, 
and that is, the distance at which the joint for the rafter must 
be located from the end of the tie beam, in order that the 
thrust of the rafter may not split oif the part against which it 
presses. This may be ascertained by Kule XI., Art. 302, for 
all cases where no iron strap or bolt is used to secure the joint : 
but where these fastenings are used the abutment may be of 
any convenient length. And in using irons here, care should 
be exercised to have the surface of pressure against the iron 
of sufficient area to prevent indentation. 

8T3. — ^To obtain the dimensions of the iron suspension rods. 
By Art 310, (17), 

and T varies (Table III.) from 5000 to 17000, according to the 
diameter inversely ; for the smaller rods are stronger in pro- 
portion than the larger ones. 

Example. — ^Taking T equal 5000, then the area of the rod 
1) G {Fig. 233) requires {Art 367) to be equal to 

corresponding to 0*469 inch diameter. This rod may be half 
inch diameter. 

Another Example. — The rod FE {Fig. 233) is loaded with 
(Art. 367) 3367 pounds, therefore 

equals the area of the rod, the corresponding diameter of which 
is 0*925. This rod may be one inch diameter. 

Again, a third example; the rod C B. This rod is loaded 
with {Art. 367) 10867 pounds, therefore 



272 AMEEICA^q' HOUSE-CAKPENTEE. 

A 10867 oi^o- -u 

equals the required area of the rod, tlie diameter correspond- 
ing to which is 1*66. This rod may therefore be If inches 
diameter. 

374. — "While discussing the principles of strains in roofs and 
deducing rules therefrom, the truss indicated in Fig. 233 has 
been examined throughout. The result is as follows : rafter, 
6x9; tie beam, (6 X 6, or) 6x9; the first brace from the 
wall, 4 J X 6 inches, with an abutting surface at the lower end 
of 6 inches, and at the upper end of 8f inches ; the other 
brace, 5J x 6 inches, with an abutting surface at the lower 
end of 9J inches, and at the upper end of 14fV inches ; the 
shortest rod, ^ inch diameter ; the next, 1 inch diameter ; and 
the middle rod, If inches diameter. 

PKACTICAI. EULES AND EXAMPLES. 

For Roofs Loaded asjper Art. 356. 

375. — Rule LIU. To obtain the dimensions of the rafter 
Multiply the value of R (Table IX., Art. 376) by tlie span 
of the roof, by the length of the rafter, and by the distance 
apart from centres at which the roof trusses are placed, all in 
feet, and divide the product by the sum of twice the height 
of the roof multiplied, by the value of P, Table L, set opposite 
the kind of wood used in the tie beam, added to the difference 
of the values of C and P in said table multiplied by 1:1 times 
the length of the arc that measures the acute angle formed 
between the rafter and a vertical line, the arc having the height 
of the roof for radius (see arc B G^ Fig. 233), and the quo- 
tient will be the area of the abutting surface of the joint at 
the foot of the rafter. To the abutting surface add its half, 
and the sum will be the area of the cross section of the rafter. 



FRAMING. 273 

This rule is upon the presumption that the rafter is secured 
from flexure by the roof beams and by braces and ties at 
short intervals, as in Fig. 233. In roofs where the rafter does 
not extend up to the ridge of the roof but abuts against a 
horizontal straining beam (<?, Fig. 237), in the rule for rafters, 
take for the length of the rafter the distance from the foot of 
the rafter to the ridge of the roof; or, a distance equal to 
what the rafter would be in the absence of a straining beam. 
The area of cross section of the straining beam should be made 
equal to that of the rafter, as found by the rule so modified. 

Example. — Find the dimensions of a rafter for a roof truss 
whose span is 52 feet, and height 13 ; the length of the rafter 
being 29 feet, the trusses placed 10 feet apart from centres, 
and the arc measuring the angle at the head of the rafter 
(having the height of the roof for radius) being 14^ feet, 
white pine being used in the tie beam. The height of this 
roof being in proportion to the span as 1 to 4, the value of It 
in Table IX. is 52*6 ; multiplying this, in accordance with the 
rule, by 52, the span of the roof, and by 29, the length of the 
rafter, and by 10, the distance between the roof trusses, the 
product is T93208. The value of P for white pine in Table L 
is 300 ; multiplying this by 2 x 13 = 26, twice the height of 
the roof, tlie product is 7800. The value of C for white pine, 
(Table I.) is 1200, hence the difference of the values of G and 
P is 1200-300 = 900; this multiplied by li, and by 14i, 
the length of the arc, the product is 16031 ; this added to the 
7800 aforesaid, the sum is 23831. The aforesaid product of 
793208, divided by this 23831, the quotient, 33*3, equals the 
area in inches of the abutting surface of the joint at the tie 
beam. To this add 16*7, its half, and the sum, 50, equals the 
area of cross section of the rafter. This divided by the thick- 
ness of the rafter, say 6 inches, the quotient, 8§, is the breadth. 
The rafter is therefore to be 6 X 8§ inches. It may be made 
6x9, avoiding the fractions. 

35 



274 



ATVTEEICAJT HOJSE-CARPENTEE. 



376. — ^The following table, calculated upon data in Art 358. 
presents the weight per foot for roofs of various inclinations, 
and covered with slate. 



TABLE IX. 





The vertical sti-ain per foot of surface supported, measured horizontally, 


When height of roof 




is to span as 








on rafters = E = 


on braces = Q =• 


1 to 8 


48 pounds 


49 '0 pounds. 


1 " 7 


48-6 " 


50-5 " 


1 " 6 


49-4 " 


51*9 


1 " 5 


50-6 " 


54- 


1 " 4 


52-6 " 


57-6 " 


1 « 3 


56-3 " 


64- 


1 *' 2 


63-7 " 


76-2 " 


1 " 1 


81- " 


101- 



To get the proportion that the height bears to the span, di- 
vide the span by the height ; then unity will be to the quotient 
as the height is to the span. In case the quotient is not a 
whole number, the required value of J? or § will not be found 
in the above table, but may be obtained thus : multiply the 
decimal part of the quotient by the difference of the values of 
i? set opposite the two proportions, between which the given 
proportion occurs as an intermediate, and subtract the product 
from the larger of the two said values of Ji ; the remainder 
will be the value of Iv required. The process is the same for 
the values of Q. 

Sample. — A roof whose span is 60 feet, has a height of 25 
feet. Then 60 divided by 25 equals 2*4. The proportion, 
therefore, between the height and span is 1 to 2*4. This pro- 
portion is an intermediate between 1 to 2 and 1 to 3. The 
values of i?, opposite these two, are 63*7 and 56'3. The dif- 
ference between these values is 7*4 ; this multiplied by 0*4, the 
decimal portion of the quotient, equals 2-96 ; this subtracted 
from 63'7, the larger value of ^, the remainder, 60*74, is the 
required value of i?. 



FEAMING. 275 

The values of E and Q are those for a roof covered with 
slate weighing Y pounds per superficial foot of the roof sur- 
face. When the roof covering is either lighter or heavier, sub- 
tract from or add to the table values, the iliiFerence of weight 
between 7 pounds and the weight of tlie covering used, and 
the remainder, or sum, will be the vahie of i? or Q required. 

377. — Rule LIY. To obtain the dimensions of braces. 
Multiply the value of Q (Table IX., Art. 376) by the distance 
apart in feet at which the roof trusses are placed, and by the 
horizontal distance in feet from a point half-way to the next 
point of support of the rafter on one side of the brace, to a 
corresponding point on the other side. The product will be 
the weight in pounds sustained at the head of the brace. To 
this add the vertical strain {Art. 360) on the suspension rod 
located at the head of the brace, and make a vertical line 
dropped from the head of the brace, as Fe^ Fig. 233, equal, 
by any convenient scale, to this sum, and draw the parallelo- 
gram Ff eg. Then Ff.^ measured by the same scale, equals 
the pressure in the axis of the brace F B. Multiply this pres- 
sure in pounds by the square of the length of the brace in feet, 
and divide the product by the breadth of the brace in inches 
multiplied by the value of ^ (Table II., Art. 2Q3). The cube 
root of the quotient will be the thickness of the brace in 
inches. If this cube root should exceed the hreadth of the 
brace, the result is not correct, and the calculation will have to 
be made anew, taking a larger dimension for the breadth. 

Exarajple. — The brace F B {Fig. 233) is of white pine, and 
IS required to sustain a pressure in its axis of T'OOO pounds 
{Art. 362). Tlie length of the brace is 12 feet and its breadth 
6 inches, what must be its thickness ? Here 7000, the pressure, 
multiplied by 144, the square of the lengtli, equals 1008000. 
The value of B is 1175 ; this by 6, the breadth of the brace, 
equals 7050. The product 1008000 divided by the product 
7050 equals 143. the cube root of which, 5'23, is the required 



276 ^IVIEEICAN HOTJSE-CAEPEXTEK. 

thickness of the brace in inches. The brace will therefore be 
5-23 by 6 inches, or 5i by 6. 

378. — Rule LY. To obtain the area of the abutting sur- 
face of the ends of braces. Divide the number of degrees 
contained in the complement of the angle of inclination by 90, 
and multiply the quotient by the difference of the values of G 
and P^ set opposite the kind of wood in the tie beam or rafter, 
in Table I., Art. 292 ; and to the product add the said value 
of P, and by the sum divide the pressure in the axis of the 
brace, and the quotient will be the area of the abutting sur- 
face. 

The complement of the angle of inclination referred to is, 
for the foot of the brace, the acute angle contained between 
the brace and a vertical line; and for the head of the, brace, 
the acute angle contained between the brace and a line per- 
pendicular to the rafter. 

Examiple. — ^To find the abutting surface of the ends of the 
brace F B (Fig. 233). The complement of the angle of incli- 
nation, for ih^foot of the brace, is that contained between the 
lines FB and F F^ and measures by the protractor, 45°. The 
tie beam is of white pine, and the values of P and C for this 
wood are 300 .and 1200 respectively, and the pressure in the 
axis of the brace is 7000 pounds. E'ow by the rule, 45 di- 
vided by 90 equals 0-5, this by the 900, the difference of the 
values of C and P, equals 450 ; to this add 300, the value of 
P, and the sum is 750. The pressure in the axis of the brace, 
7000, divided by this 750, equals 9§, the required area of the 
abutting surface at the foot of the rafter. The complement of 
the angle of inclination for the head of the brace is that con- 
tained between the lines ^ P and Pjp, and measures by the 
protractor 19°. The rafter being of white pine, the values of 
P and C are as before. By the rule, 19 divided by 90 equals 
0*2^, and this multiplied by 900, the difference of the values 
of P and 6> equals 190 ; to this add 300, the value v-f P, and 



FRAMING. 277 

the sum is 490. The pressure, 7000, divided by this 490, 
equals 14*3 inches, the required area of the abutting surface at 
the head of the brace. 

379. — To obtain the dimensions of the tie beam. Tie beams 
are subjected to two kinds of strain — tensile and transverse. 

Rule LYI. — To guard against the tensile strain, multiply the 
value of B, (Table IX., Art. 376) by three times the distance 
apart at which the trusses are placed, and by the square of the 
span of the truss, both in feet. Divide this product by the 
value of T, (Table III., Art. 308) set opposite the kind of wood 
in the tie beam, multiplied by 8 times the height of the roof 
in feet, and by the breadth of the tie beam in inches. The 
quotient will be the required depth in inches. 

The result thus obtained is usually smaller than that re- 
quired to resist the cross strain to which the tie beam is sub- 
jected. The dimensions required to resist this strain, where 
there is simply the weight of the ceiling to support, may be 
obtained by this rule : 

Rule LYII. — Multiply the cube of the longest unsupported 
part of the tie beam by 400 times the distance apart at which 
the trusses are placed, both in feet ; and divide the product by 
the breadth of the tie beam in inches, multiplied by the value 
of E^ (Table II., Art. 293) set opposite the kind of w^ood in the 
tie beam, and the cube root of the quotient will be the re- 
quired depth of the tie beam in inches. 

The result thus obtained may not be sufficient, in some cases, 
to resist the strains to which the tie beam is subjected in the 
hands of the workmen during the process of framing. 

Rule LYIII. — To resist these strains the area of cross sec- 
tion in inches should be at least equal to the length in feet. 

Example. — ^The tie beam in Fig. 233. For this case we have 
the value of R 52*6, the trusses placed 10 feet from centres, 
the span 52 feet, the height 13 feet, the breadth 6 inches, and 
<.lie value of T 2367. Then by the rule, 52-6 x 3 x 10 x 52* 



278 AMERICAN HOrSE-CAEPENTEE. 

=:: 4266912, and 236T x 8 x 13 x 6 = 1477008; the former 
product divided by the latter, the qnotient equals 2'9, equals 
the required depth of the tie beam in inches. The other 
strains will require the depth to be more. To resist the cross 
strains, we have the longest unsupported part of the tie beam 
8f feet, (this dimension is frequently greater than this.) distance 
from centres 10 feet, and breadth 6 inches. Then, by the rule, 
8|3 X 400 X 10 ^ 2603852, and 6 x 1750 = 10500 ; the former 
product divided by the latter, the quotient is 248, the cube root 
of which, 6*28, equals the required depth in inches. The tie 
beam therefore is to be 6 by 6-28 inches, or 6 x 7 inches. But 
if not guarded against severe accidental strains from careless 
handling this size would be too small. It would, in this case, 
require to be 52 inches area of cross section, say 6x9 inches. 

380. — To obtain the diameter of the suspension rods, when 
.made of s^ood wrous^ht iron. 

Rule LIX. — Divide the weight or vertical strain, in pounds, 
by 4000. The square root of the quotient will be the required 
diameter of the rod in inches. 

Example. — A suspension rod is required to sustain 16000 
pounds, what must be its diameter ? Dividing by 4000, the 
quotient is 4 ; the square root of which, 2, is the required 
diameter. 

The vertical strain on any rod is equal to the weight of so 
much of the ceiling as is supported by the rod, added to the 
vertical strain caused by each brace that is footed in the tie 
beam at the rod. The weight of the ceiling supported by a 
rod, is equal to ten times the distance apart in feet at which 
the trusses are placed, multiplied by half the distance in feet 
between the two next points of support, one on either side of 
the rod. Tlie vertical strain caused by the braces can be as- 
certained geometrically, as in Art. 360. 

381. — When the suspension rods are located as in Fig. 233, 
dividing the span into equal parts, the diameter ol the rods 



FRAMING. 279 

may be obtained without the preliminaiy calculation of the 
strain, as follows : 

Rule LX. — For the first rod from the wall. Multiply the 
distance apart at which the trusses are placed by the distance 
apart between the suspension rods, and divide the product by 
400. The square root of the quotient will be the required 
diameter of the rod. 

Example. — Rod D G^ Fig. 233. In this figure the rods are 
located at 8f feet apart, and the distance between the trusses 
is 10 feet. Therefore, 10 x 8| == 86i ; this divided by 400, 
the quotient is 0*2167, the square root of which, 0*4:65, is the 
required diameter. The diameter may be half an inch. 

Hide LXL — For the second rod from the wall. To the 
value of Q (Table IX., Art. 376) add 20, and multiply the sum 
by the distance apart at which the trusses are placed and by 
the distance between the rods, both in feet, and divide the pro- 
duct by 8000. The square root of the quotient will be the re- 
quired diameter. 

Example. — Rod F E^ Fig. 233. The distances apart in this 
case are as stated in last example. The value of Q is 57*6, and 
when added to 20 equals 77-6. Therefore, 77-6 X 10 x 8| = 
6673J ; this divided by 8000, the quotient is 0-8341, the square 
root of which, 0*91, is the required diameter. This rod may 
be one inch diameter. 

Hule LXIL — ^For the centre rod. To the value of Q (Table 
IX., Art 376) add 5, and multiply the sum by the distance 
apart at which the trusses are placed and by the distance apart 
between the rods, both in feet, and divide the product by 2000. 
The sqnararoot of the quotient will equal the required diameter. 

Example. — Rod C B^ Fig. 233. The distances apart as be- 
fore, and the value of Q the same. To Q add 5, and the sum 
is 62-6. Then 62*6 x 10 x 8f = 5425J ; this divided by 2000, 
the quotient is 2*7126, the square root of which, 1*647, equal? 
the required diameter. • This rod may be If inches diameter. 



280 ♦ iLMEEICAN HOUSE-CAEPENTEK. 

382. — For all wrought iron straps and bolts the dimensions 
may be found by this rule. 

Rule LXIII. — Divide the tensile strain on the piece, in 
pounds, by 5000, and the quotient will be the area of cross 
section of the required bar or bolt, in inches. 

383. — Roof-beams, jack-rafters, and purlins. All pieces of 
timber subject to cross strains will sustain safely much greater 
strains when extended in one piece over two, three, or more 
distances between bearings ; therefore roof-bea,ras, jack-rafters, 
and purlins should, if possible, be made in as long lengths as 
practicable ; the roof-beams and purlins laid on, not framed 
into, the principal rafters, and extended over at least two 
spaces, the joints alternating on the trusses ; and likewise the 
jack-rafters laid on the purlins in long lengths. The dimen- 
sions of these several pieces may be obtained by the following 
rule : 

Rule LXIY.— From the value of Q (Table IX., AH. 376) 
deduct 10, and multiply the remainder by 33 times the distance 
from centres in feet at which the pieces are placed, and by the 
cube of the distance between bearings in feet ; divide the pro- 
duct by the value of E (Table II., Art. 293) for the kind of 
wood used and extract the square root of the quotient. The 
square root of this square root will be the required depth in 
inches. Multiply the depth thus obtained by the decimal 0-6, 
and the product will be the required breadth in inches. 

Example, — ^Roof-beams of white pine placed 2 feet from cen- 
tres, resting on trusses placed 10 feet from centres, the height 
and the span of the roof being in proportion as 1 to 4. In 
this case the value of Q is 57*6. By the rule, 57*6 — 10 = 47-6, 
and 47-6 x 33 x 2 x 10^ =r 3141600. This iivided by 1750, 
the quotient is 1795*2, the square root of which is 42*37, and the 
square root of 42*37 is 6*5, the required depth. This multi- 
plied by 0*6 equals 3*9, the required breadth. These roof 
beams may therefore be 4 by 6-| inches. 



FRAMING. 



281 



38 i. — Five examples of roofs are shown at Figs. 234, 235, 

236, 237, and 238. In Fig. 234, a is an iron suspension rod, 

5 h are braces. In Fig. 235, 

(2, «, and 5 are iron rods, and 

dd^cG^ are braces. In Fig. 

236, ah are iron rods, (^(^ i' 

i 
braces, and c the straining - 

beam. In Fig. 237, aa^hl^ 






are iron rods, ee^ dd^ are braces, and c is a straining beam. 
In Fig. 238, purlins are located at PP, &c. ; the inclined beam 
that lies upon them is the jack-rafter; the post at the ridge is 
the king post, the others are queen posts. In this design the tie 
beam is increased in height along the middle by a strengthen- 
ing piece {Art. 348), for the purpose of sustaining additional 
weight placed in the room formed in the truss. 

385. — Fig. 239 shows a method of constructing a truss having 
a huilt-rih in the place of principal rafters. The proper form 

for the curve is that of a parabola, {Art 127.) This curve, 

3f5 



282 



AMERICAN HOUSE-CARPENTEE. 



when as flat as is described in the figure, approximates so 
near to that of the circle, that the latter may be used in its 
stead. The height, a 5, is just half of a c, the curve to pass 

through the middle of the rib. 
The rib is composed of two series 
of abutting pieces, bolted toge- 
ther. These pieces should be as 
long as the dimensions of the tim- 
ber will admit, in order that there 
may be but few joints. The sus- 
pending pieces are in halves, 
notched and bolted to the tie- 
beam and rib, and a purlin is 
framed upon the upper end of 
each. A truss of this construc- 
tion needs, for ordinary roofs, no 
^ }=^ diagonal braces between the sus- 
pending pieces, but if extra 
strength is required the braces 
may be added. The best place 
for the suspending pieces is at the 
joints of the rib. A rib of this 
kind will be sufficiently strong, 
if the area of its section contain 
about one-fourtli more timber, 
than is required for that of a raf- 
ter for a roof of the same size. 
The proportion of the depth to 
the thickness should be about aa 
10 is to 7. 
386 — Some writers have given designs for roofs similar to 
Fig. 240, having the tie-beam omitted for the accommodation 
of an arch in the ceiling. This and all similar designs are se- 




FKAMINU. 



283 



riously objectionable, and should always be avoided ; as the 
small height gained by the omission of the tie-beam can never 




Fig. 240. 



compensate for the powerful lateral strains, which are exerted 
by the oblique position of the supports, tending to separate the 



284 



AMERICAN HOrSE-CAEPENTER. 



walls. Where an arcli is required in the ceiling, the best plan 
is to carry np the walls as high as the top of the arch. Then, 
by using a horizontal tie-beam, the oblique strains will be en- 
tirely removed. Many a public building, by my own obser- 
vation, has been all but ruined by the settling of the roof, 
consequent upon a defective plan in the formation of the truss 
in this respect. It is very necessary, therefore, that the hori- 
zontal tie-beam be used, except where the walls are made so 
strong and firm by buttresses, or other support, as to prevent 
a possibility of their separating. 




n g 
Fig. 241. 

387. — "Fig. 241 is a method of obtaining the proper lengths and 
bevils for rafters in a hip-poof : a h and h c are walls at the angle 
of the building ; 5 e is the seat of the hip-rafter and gf of a 
jack or cripple rafter. Draw e A, at right angles to h e^ and make 
it equal to the rise of the roof; join 5 and A, and h h will be the 
length of the hip-rafter. Through e, draw d t, at right angles to 
he J upon b, with the radius,- h A, describe the arc, h i, cutting ^« 
in i / join h and ^, and extend ^y to meet himj / then gj will 



FRAMING. 



285 



be the length of the jack-rafter. The length of each jack-rafter is 
found in the same manner — by extending its seat to cut the line, 
b i. From/, draw fk, at right angles io fg, also/Z, at right 
angles to be; make/ A; equal to // by the arc, I k, or make g- k 
equal to g j by the arc, j k ; then the angle at j will be the top- 
bevil of the jack-rafters, and the one at k will be the down-bevil* 
388. — To find the backing of the hip-rafter. At any con- 
venient place in b e, {Fig. 241,) as o, draw m n, at right angles to 
be; from o, tangical to b h, describe a semi-circle, cutting 6 e in 
s ; join m and s and n and s ; then these lines will form at s the 
proper angle for beviling the top of the hip-rafter. 

DOMES.t 




Fig. 242. 




Fig. 248. 

* The lengths and bevils of rafters for roof-valleys can also be found by he aDoff 
process t See also Art. 237. 



2S6 



AMERICAN HOUSE-CARPEXTER. 



3S9. — The most usual form for domes is that of the sphere, the 
base bemg circular. When the interior dome does r ot rise too 
high, a horizontal tie may be thrown across, by which any de- 
gree of strength required may be obtained. Fig. 242 shows a 
section, and Fig. 213 the plan, of a dome of this kind, a h being 
the tie-beam in both. Two trusses of this kind, {Fig. 212,) ]^)a- 
rallel to each other, are to be placed one on each side of the open- 
ing in the top of the dome. Upon these the whole framework is to 
depend for support, and their strength must be calculated accord- 
ingly. (See the first part of this section, and Art. 356.) If the 
dome is large and of importance, two other trusses may be intro- 
duced at right angles to the foregoing, the tie-beams being pre- 
served in one continuous length by framing them high enough to 
pass over the others. 




Fig. 244. 




390. — When the interior dome rises too hieh to admit of a leva. 



FRAMING. 287 

tie-beam, the framing may be composed of a succession of ribs 
standing upon a continuous circular curb of timber, as seen at 
Fig. 244 and 245; — the latter being a plan and the former a sec- 
tion. This curb must be well secured, as it serves in the place 
of a tie-beam to resist the lateral thrust of the ribs. In small 
domes, th^se ribs may be easily cut from wide plank ; but, where 
an extensive structure is required, th^y must be built in two 
thicknesses so as io break joints^ in the same manner as is descri- 
bed for a roof at Art. 885. They should be placed at about two 
feet apart at the base, and strutted as at a in Fig. 244. 

391. — The scantling of each thickness of the rib may be as 
follows : 

For domes of 24 feet diameter, 1x8 inches. 
" " 36 " 1^x10 " 

" ' 60 " 2x13 " 

"• 90 " 2^x13 " 

« " 108 " 3x13 " 

392. — Although the outer and the inner surfaces of a dome 
may be finished to any curve that may be desired, yet the framing 
should be constructed of such a form, as to insure that the curve 
of equilibrium will pass through the middle of the depth of the 
framing. The nature of this curve is such that, if an arch or 
dome be constructed in accordance with it, no one part of the 
structure will be less capable than another of resisting the strains 
and pressures to which the whole fabric may be exposed. The 
curve of equilibrium for an arched vault or a roof, where the load 
is equally diffused over the whole surface, is that of a parabola, 
{Art. 127 ;) for a dome, having no lantern, tower or cupola above 
it, a cubic parabola, {Fig. 246 ;) and for one having a tower, (fcc, 
above it, a curve approaching that of an hyperbola must be adopted, 
as the greatest strength is required at its upper parts. If the 
curve of a dome be circular, (as ir. the vertical section. Fig. 244,) 
ihe pressure will have a tendency to burst the dome outwards at 
about one-third of its height. Therefore, when this form is used 



288 



AMERICAN HOUSE- CARPENTER. 



in the construction of an extensive dome, an iron band should be 
placed around the framework at that height ; and whatever may 
be the form of the curve, a bond or tie of some kind is necessary 
around or across the base. 

If the framing be of a form less convex than the curve of 
equilibrium, the weight will have a tendency to crush the ribs in- 
wards, but this pressure may be effectually overcome by strutting 
between the ribs ; and hence it is important that the struts be so 
placed as to form continuous horizontal circles. 








^^^-^ 




(' 


/^ 









/ 


/ 











/ 













/ 




1 




n/ 






1 1 




A 















a j i h 



Fig. 246. 



393. — To describe a cubic parabola. Let a b, {Fig. 246,) be 
the base and b c the height. Bisect a b aX d, and divide a d into 
100 equal parts; of these give d e 26. ef 18^,/ g 14|, g h 12^, 
h i lOf, ij 9ij and the balance, 8f, toj a; divide b c into 8 equal 
parts, and, from the points of division, draw lines parallel to a b. 
to meet perpendiculars from the several points of division in a b, 
at the points, o, o, o, &c. Then a curve traced through these 
points will be the one required. 

394. — Small domes to light stairways, &c., are frequently made 
elliptical in both plan and section ; and as no two of the ribs in 
one quarter of the dome are alike in form, a method for obtaining 
the curves is necessary. 

395. — To find the curves for the ribs of an elliptical dome 
Let a bed, {Fig. 247,) be the plan of a dome, and e f the seat 



FRAMING. 



289 




Fig. 247. 

ot one of the ribs. Then take e/for the transverse axis ana 
twice the rise, o g, of the dome for the conjugate, and describe, 
(according to Art. 115, 116, &c.,) the semi-ellipse, e g f, which 
will be the curve required for the rib, e g f. The other ribs are 
found in the same manner. 



b 4 




Fig. 248. 



396. — To find the shape of the covering for a spherical 
dome. Let ,4, [Fig. 248,) be the plan and B the section of a 
given dome. From «, draw a c, at right angles to a b ; find the 
stretch-out, {Aj^t. 92,) of o b, and make d c equal to it ; divide the 
arc, bj and the line, d c, each into a like number of equal parts^ 

37 



290 



AMERICAN HOUSE-CARPENTER. 



as 5, (a large number .vill insure greater accuracy than a small 
one ;) upon c, through the several points of division in c d^ describe 
the arcs, o c? o, 1 e 1, 2/ 2, &c. ; make d o equal to half the width 
of one of the boards, and draw o s, parallel to a c ; join 5 and a, 
and from the points of division in the arc, o b, drop perpendicu- 
lars, meeting a 5 in ij k I ; from these points, draw i 4, j 3, &c., 
parallel to a c; make d o, e 1, (fee, on the lower side of a c, equal 
to d 0, e 1, 6cc., on the upper side ; trace a curve through the 
points, 0, 1, 2, 3, 4, c, on each side o{ d c ; then o c o will be 
the proper shape for the board. By dividing the circumference of 
the base, A, into equal parts, and making the bottom, o d o,of the 
board of a size equal to one of those parts, every board may be 
made of the same size. In the same manner as the above, the 
shape of the covering for sections of another form may be found, 
such as an ogee, cove, &c. 




397. — To find the curve of the boards lohen laid in horizon- 
tal coiirses. Let ABC, {Fig. 249,) be the section of a given 
dome, and D B its axis. Divide B C into as many parts as 
there are to be courses of boards, in the points, 1, 2, 3, &c. ; 
through 1 and 2, draw a line to meet the axis extended at a / 
then a will be the centre for describing the edges of the board; F. 
Through 3 and 2, draw 36; then b will be the centre for describing 
F. Through 4 and 3, draw 4 d; then d will be the centre for G. 
B is tiie centre for the arc, 1 o. If this method is taken t.^ tno. 



FRAMING. 



291 



the centres for the boards at the base of the dome, they would 
occur so distant as to make it impracticable : the following metl. od 
is preferable for this purpose. G being the last board obtained by 
the above method, extend the curve of its inner edge until it 
merts the axis, D B^in e ; from 3, through e, draw 3/, meeting 
the arc, A B, in/; join /and 4, /and 5 and/ and 6, cutting the 
axis, D B, in s, n and m ; from 4, 5 and 6, draw lines parallel to 
A Cand cutting the axis in c,p and r ; make c 4, {Fig, 250,) 




equal to e 4 in the previous figure, and c 5 equal to c 5 also in the 
previous figure ; then describe the inner edge of the board, H^ 
according to Art. 87 : the outer edge can be obtained by gauging 
from the inner edge. In like manner proceed to obtain the next 
board — taking p 5 for half the chord and p n for the height of the 
segment. Should the segment be too large to be described 
easily, reducf; it by finding intermediate points in the curve, as at 
Art, 86. 




398. — 7b find the shape of the angle-rih for a polygonal 
dcme. Let A G H, {Fig. 251,) be the plan of a given dome, and 



292 



AMERICAN HOUSE-CARPENTER. 



C DsL vertical section taken at the line, ef. From 1, 2, 3, &c.. 
in the arc. C D, draw ordinates, parallel to A D, to meety" G ; 
from the points of intersection on / G, draw ordinates at right- 
angles to/ G ; make s 1 equal to o 1, 5 2 equal to 2, &c. ; then 
GfB, obtained in this way, will be the angle-rib required. The 
best position for the sheathing-boards for a dome of this kind is 
horizontal, but if they are required to be bent from the base to 
the vertex, their shape may be found in a similar manner to that 
shown at Fig, 248. 

BRIDGES. 

399. — Various plans have been adopted for the construction of 
bridges, of which perhaps the following are the most useful. 
Fig. 252 shows a method of constructing wooden bridges, where 
the banks of the river are high enough to permit the use of the 
tie-beam, a b. The upright pieces, c d, are notched and bolted 
on in pairs, for the support of the tie-beam. A bridge of this 
construction exerts no lateral pressure upon the abutments. This 
method may be employed even where the banks of the river are 
low, by letting the timbers for the roadway rest immediately upon 
the tie-beam. In this case, the framework above will serve the 
purpose of a railing. 




Fig. 252. 



400. — Fig. 253 exhibits a wooden bridge without a tie-beam. 
Where staunch buttresses can be obtained, this method may be 
recommended ; but if there is any doubt of their stability, it 



FRAMING. 



293 




Fig. 253. 



should not be attempted, as it is evident that such a system ol 
framing is capable of a tremendous lateral thrust. 




Fi§. 254. 



401. — Fi^. 254 represents a wooden bridge in which a built-rib, 
(see Art. 385,) is introduced as a chief support. The curve of 
equilibrium will not differ much from that of a parabola : this,, 
therefore, may be used — especially if the rib is made gradually a 
little stronger as it approaches the buttresses. As it is desirable 
that a bridge be kept low, the following table is given to show the 
least rise that may be given to the rib. 



Span in feet. 


Least rise in feet. 


iSpan in feet. 


Least rise in feet. 


Span in feet. 


Least rise in feet. 


30 


0-5 


120 


7 


280 


24 


40 


0-8 


140 


8 


300 


28 


50 


1-4 


160 


10 


320 


32 


60 


2 


180 


11 


3e50 


39 


70 


2i 


200 


12 


380 


47 


80 


3 


220 


14 


400 


53 


90 


4 


240 


17 






100 


5 


260 


20 







The rise should never be made less than this, but in all cases 



294 



AMERICAN HOUSE-CARPENTER. 



greater if practicable ; as a small rise requires a greater quantity 
of timber to make the bridge equally strong. The greatest uni- 
form Aveight with which a bridge is likely to be loaded is, proba- 
bly, that of a dense crowd of people. This may be estimated at 
66 pounds per square foot, and the framing and gravelled road- 
way at 234 pounds more ; which amounts to 300 pounds on a 
square foot. The following rule, based upon this estimate, may 
be useful in determining the area of the ribs. Hide LXY. — 
Multiply the width of the bridge by the square of half the span, 
both in feet ; and divide this product by the rise in feet, multi- 
plied by the number of ribs ; the quotient, multiplied by the 
decimal, 0*0011, will give the area of each rib in feet. When 
the roadway is only planked, use the decimal, 0*0007, instead of 
O'OOll. Example. — What should be the area of the ribs for a 
bridge of 200 feet span, to rise 15 feet, and be 30 feet wide, with 
3 curved ribs ? The half of the span is 100 and its square is 
10,000 ; this, multiplied by 30, gives 300,000, and 15, multi- 
plied by 3, gives 45 ; then 300,000, divided by 45, gives 6666|, 
which, multiplied by O'OOll, gives 7*333 feet, or 1056 inches for 
the area of each rib. Such a rib may be 24 inches thick by 44 
inches deep, and composed of 6 pieces, 2 in width and 3 in depth. 




Fig. 255. 



402. — The above rule gives the area of a rib, that would be re- 
quisite to support the greatest possible miiform load. But ir. 
large bridges, a variahle load, such as a heavy wagon, is capable 
of exerting much greater strains ; in such cases, therefore, the 
rib should be made larger. The greatest concentrated load a 



FRAMING. 295 

bridge will be likely to encounter, may be estimated at from alout 
20 to 50 thousand pounds, according to the size of the bridge. 
This is capable of exerting the greatest strain, when placed al 
about one-third of the span from one of the abutments, as at b 
{Fig. 255.) The weakest point of the segment, 6 ^ c, is at gj 
the most distant point from the chord line. The pressure exerted 
at b by the above weight, may be considered to be in the direction 
of the chord lines, b a and be; then, by constructing the paral- 
lelognT. of forces, e bf d, according to Art. 258, bf will show 
the pressure in the direction, b c. Then the scantling for the rib 
may be found by the following rule. 

Hule LXVl. — Multiply the pressure in pounds in the direc- 
tion b c, by the distance g h, and by the square of the distance 
b c, both in feet ; and divide the product by the united breadth 
in inches of the several ribs, multiplied by the value of B, 
(Table IL, A7't. 293) for the kind of wood used ; and the cube 
root of the quotient will be the required depth of the rib in 
inches. 

Exarrhple. — A bridge is to have three white pine ribs each 
20 inches wide ; the pressure in the direction b c, (Fig. 255) 
is equal to 60,000 pounds, the distance b c equals 60 feet, and 
the distance g h equals 10 feet. What must be the depth of 
the ribs, the value of B (Table II.) being for white pine 1175 ? 
Here, by the rule, 60,000 X 10 x 60'' = 2,160,000,000. Then 
1175 X 3 X 20 = 70,500. The former product divided by the 
latter equals 30,638, the cube root of which, 31*29, equals 
the required depth in inches. The ribs are, therefore, to be 20 
by 31^ inches. 

403. — In constructing these ribs, if the span be not over 50 
feet, each rib may be made in two or three thicknesses of timber, 
(three thicknesses is preferable,) of convenient lengths bolted 
toge:;lier ; but, in larger spans, where the rib will be such as to 
render it difficult to procure timber of sufficient breadth, they 
may be constructed by bending the pieces to the proper curve, 



295 



AMERICAN HOUSE-CARPENTER. 



and bolting them together. In this case, where timber of suffi 
cient length to span the opening cannot be obtained and scarfing 
is necessary, such joints must be made as will resist both tension 
and compression, (see Fig. 264 ) To ascertain the greatest depth 
for the pieces which compose the rib, so that the process of bend 
ing may not injure their elasticity, multiply the radius of curvature 
in feet by the decimal, 0-05, and the product will oe the depth m 
inches. Example. — Suppose the curve of the rib to be described 
with a radius of 100 feet, then what should be the depth ? The 
radius in feet, 100, multiplied by 0*05, gives a product of 5 inches. 
White pine or oak timber, 5 inches thick, would freely bend to 
the above curve ; and, if the required depth of such a rib be 2l 
inches, it would have to be compos-ed of at least 4 pieces. Pitch 
pine is not quite so elastic as white pine or oak — its thickness 
may be found by using the decimal, 0*046, instead of 0*05. 




Fig. 256. 



404. — When the span is over 250 feet, b, framed rib, formed as 
in Fig. 256, would be preferable to the foregoing. Of this, the 
upper and the lower edges are formed as just described, by bend- 
ing the timber to the proper curve. The pieces that tend to the 
centre of the curve, called radials, are notched and bolted on in 
pairs, and the cross-braces are halved together in the middle, and 
abut end to end between the radials. The distance between the 
ribs of a bridge should not exceed about 8 feet. The roadway 



FRAMING. 297 

should be supported by vertical standards bolted to the ribs at 
about every 10 to 15 feet. At the place where they rest on the 
ribs, a double, horizontal tie should be notched and bolted on the 
back of the ribs, and also another on the under side ; and diago- 
nal braces should be framed between the standards, over the space 
between the ribs, to prevent lateral motion. The timbers for the 
roadway may be as light as their situation will admit, as all use- 
less timber is only an unnecessary load upon the arch. 

405. — It is found that if a roadway be 18 feet wide, two car- 
riages can pass one another without inconvenience. Its width 
therefore, should be either 9, 18, 27 or 36 feet, according to the 
amount of travel. The width of the foot-path should be 2 feet 
for every person. When a stream of water has a rapid current, 
as few piers as practicable should be allowed to obstruct its 
course ; otherwise the bridge will be liable to be swept away by 
freshets. When the span is not over 300 feet, and the banks of 
the river are of sufficient height to admit of it, only one arch 
should be employed. The rise of the arch is limited by the form 
of the roadway, and by the height of the banks of the river 
(See Art. 401.) The rise of the roadway should not exceed one 
in 24 feet, but, as the framing settles about one in 72, the roadway 
should be framed to rise one in 18, that it may be one in 24 after 
settling. The commencement of the arch at the abutments — the 
spring, as it is termed, should not be below high-water mark : 
and the bridge should be placed at right angles with the course of 
the current. 

406. — The best material for the abutments and piers of a 
bridge, is stone ; and, if possible, stone should be procured for the 
purpose. The following rule is to determine the extent of the 
abutments, they being rectangular, and built with stone weighing 
120 lbs. to a cubic-foot. Idule LXYII. — Multiply the square 
of the height of the abutment by 160, and divide this product by 
the weight of a square foot of the arch, and by the rise of the arch ; 
add unity to the quotient, and extract the square-root. Diminish 
the square-root by unity, and multiply the root, so diminished, by 



298' AMERICAN HOUSE-CARPENTER. 

half the span of the arch, and by the weight of a square-foot ol 
the arch. Divide* the last product by 120 times the height of the 
abutment, and the quotient will be the thickness of the abutment, 
Examj^le. — Let the height of the abutment from the base to the 
springing of the arch be 20 feet, half the span 100 feet, the weight 
of a square foot of the arch, including the greatest possible load 
upon it, 300 pounds, and the rise of the arch 18 feet — what should 
be its thickness ? The square of the height of the abutment, 
400, multiphed by 160, gives 64,000, and 300 by 18, gives 5400 ; 
64,000, divided by 5400, gives a quotient of 11-852, one added to 
this makes 12-852, the square-root of which is 3'6 ; this, less one, 
is 2*6 ; this, multiplied by 100, gives 260, and this again by 300, 
gives 78,000 ; this, divided by 120 times the height of the abut- 
ment, 2400, gives 32 feet 6 inches, the thickness required. 

The dimensions of a pier will be found by the same rule. 
For, although the thrust of an arch may be balanced by an ad- 
joining arch, when the bridge is finished, and while it remains 
uninjured ; yet, during the erection, and in the event of one arch 
being destroyed, the pier should be capable of sustaining the en- 
th'e thrust of the other. 

407. — Piers are sometimes constructed of timber, their princi- 
pal strength depending on piles driven into the earth, but such 
piers should never be adopted where it is possible to avoid them ; 
for, being alternately wet and dry, they decay much sooner than 
the upper parts of the bridge. Spruce and elm are considered 
good for piles. Where the height from the bottom of the 
river to the roadway is great, it is a good plan to cut them off at 
a little below low-water mark, cap them with a horizontal tie. 
and upon this erect the posts for the support of the roadway. 
This method cuts off the part that is continually wet from that 
which is only occasionally so, and thus affords an opportunity for 
replacing the upper part. The pieces which are immersed will 
last a great length of time, especially when of elm ; for it is a 
well-established fact, that timber is less durable when subj(;ct to 



FRAMING. 



299 



ilteriiate dryness and moisture, than when it is either continually 
wet or continually dry. It has been ascertained that the piles 
luider London bridge, after having been driven about 600 years, 
yere not materially decayed. These piles are chiefly of elm, and 
vhoUy immersed. 




Fig. 257. 



408. — Centres for stone bridges. Fig. 257 is a design for a 
centre for a stone bridge where intermediate supports, as piles 
driven into the bed of the river, are practicable. Its timbers are 
so distributed as to sustain the weight of the arch-stones as they 
are being laid, without destroying the original form of the centre ; 
and also to prevent its destruction or settlement, should any of the 
piles be swept away. The most usual error in badly-constructed 
centres is, that the timbers are disposed so as to cause the framing 
to rise at the crown, during the laying of the arch-stones up the 
sides. To remedy this evil, some have loaded the crown with 
heavy stones ; but a centre properly constructed will need no 
such precaution. 

Experiments have shown that an arch-stone does not press 
upon the centring, until its bed is inclined to the horizon at an 
angle of from 30 to 45 degrees, according to the hardness of the 
stone, and whether it is laid in mortar or not. For general pur- 
poses, the point at which the pressure commences, may be con- 
sidered to be at that joint which forms an angle of 32 degrees 
with the horizon. At this pomt, the pressure is inconsiderable, 



300 



AMERICAN HOUSE-CAEPENTEE. 



but gradually increases towards tbe crown. The following 
table gives tke portion of the weight of the arch stones that 
presses upon the framing at the various angles of inclination 
formed by the bed of the stone with the horizon. The press- 
ure perpendicular to the curve is equal to the weight of the 
arch stone multiplied by the decimal 



0, when the angle of inclination is 32 degrees. 

04 " 

08 " 

12 " 

\1 " 

21 " " 

25 *' 

29 " " 

33 " ** 

3*7 " " 

4 « 

44 « 

48 « 

52 " " 

54 " " 



11 ti a 


34 


(( <( (( 


36 


« U (( 


38 


(( « ({ 


40 


« (( (( 


42 


(( « « 


44 


ft i( le 


46 


(( « (I 


48 


(( (( (I 


60 


(( « <( 


52 


« « « 


54 


« (( « 


56 


(( <( (( 


58 


« « (( 


60 




From this it is seen that at the inclination of M degrees the 
pressure equals one-quarter the weight of the stone ; at 57 de- 
grees, half the weight ; and when a vertical 
line, as a h, {.Fig. 258,) passing through the 
centre of gravity of the arch-stone, does not 
fall within its bed, c d, the pressure may be 
considered equal to the whole weight of the 
stone. This will be the case at about 60 de- 
grees, when the depth of the stone is double its breadth. The 
direction of these pressures is considered in a line with the ra- 
dius of the curve. The weight upon a centre being known, 
the pressure may be estimated and the timber calculated ac- 
cordingly. But it must be remembered that the whole weight 
is never placed upon the framing at once — as seems to have 



Fig. 258. 



FRAMING. 



301 



been the idea had in view by the designers of some centres. 
In building the arch, it chould be commenced at each buttress 
at the same time, (as is generally the case,) and each side 
should progress equally towards the crown. In designing the 
framing, the effect produced by each successive layer of stone 
should be considered. The pressure of the stones upon one 
side sliould, by the arrangement of the struts, be counterpoised 
by that of the stones upon the other side. 

409. — Over a river whose stream is rapid, or where it is ne- 
cessary to preserve an uninterrupted passage for the purposes 
of navigation, the centre must be constructed without interme- 
diate supports, and without a continued horizontal tie at the 




Fig. 259. 

base ; such a centre is shown at Fig. 259. In laying the stones 
from the base up to a and c, the pieces, h d and h d, act as ties 
to prevent any rising at h. After this, while the stones are 
being laid from a and from c to 5, they act as struts : \h.Q piece, 
fg^ is added for additional security. Upon this plan, with 
some variation to suit circumstances, centres may be con- 
structed for any span usual in stone-bridge building. 

410. — ^In bridge centres, the principal timbers should abut, 
and not be intercepted by a siispension or radial piece between. 
These should be in halves, notched on each side and bolted. 
The timbers should intersect. as little as possible, for the more 



302 A]VIERICA1T HOUSE-CARPEin^K. 

joints the greater is the settling ; and halving them together is 
a bad practice, as it destroys nearly one-half the strength of 
the timber. Ties should be introduced across, especially ^Yhere 
many timbers meet ; and as the centre is to serve but a tem- 
porary purpose, the whole should be designed with a view to 
employ the timber afterwards for other uses. For this reason, 
all unnecessary cutting should be avoided. 

411. — Centres should be suflBciently strong to preserve a 
staunch and steady form during the whole process of building ; 
for any shaking or trembling will have a tendency to prevent 
the mortar or cement from setting. For this purpose, also, 
the centre should be lowered a trifle immediately after the 
key-stone is laid, in order that the stones may take their bear- 
ing before the mortar is set; otherwise the joints will open on 
the under side. The trusses, in centring, are placed at the 
distance of from 4: to 6 feet apart, according to their strength 
and the weight of the arch. Between ev^y two trusses, diago- 
nal braces should be introduced to prevent lateral motion. 

412. — In order that the centre may be easily lowered, the 
frames, or trusses, should be placed upon wedge-formed sills ; 
as is shown at d^ {Fig- 259.) These are contrived so as to ad- 
mit of the settling of the frame by driving the wedge, tZ, with 
a maul, or, in large centres, with a piece of timber mounted as 
a battering-ram. The operation of lowering a centre should 
be very slowdy performed, in order that the parts of the arch 
may take their bearing uniformly. Tlie wedge pieces, instead 
of being placed parallel with the truss, are sometimes made 
sufficiently long and laid through the arch, in a direction at 
right angles to that shown at Fig. 259. This method obviates 
the necessity of stationing men beneath the arch during the 
process of lowering ; and was originally adopted with success 
soon after the occurrence of an accident, in lowering a centre, 
by which nine men were killed. 

413. — To give some idea of the manner of estimating the pres- 



FRAMING. 303 

snres, in order to select timber of tlie proper scantling, calculate 
{Art. 408) the pressure of the arch-stones from i to &, {Fig. 259,) 
and snppose half this pressure concentrated at a, and acting in 
the direction af. Then, by the parallelogram of forces, {Art. 
258,) the strain in the several pieces composing the frame, 
hd a^ may be computed. Again, calculate the pressure of that 
portion of the arch included between a and c, and consider 
half of it collected at 5, and acting in a vertical direction ; 
then, by the parallelogram of forces, the pressure on the beams, 
h d and h d, may be found. Add the pressure of that portion 
of the arch which is included between i and 5 to half the 
weight of the centre, and consider this amount concentrated 
at d^ and acting in a vertical direction ; then, by constructing 
the parallelogram of forces, tlie pressure upon dj may be as- 
certained. 

414. — The strains having been obtained, the dimensions of 
the several pieces in the frames had and h c d^ may be found 
by computation, as directed in the case of roof trusses, from 
Arts. 375 to 380. The tie-beams hd^hd^ if made of sufficient 
size to resist the compressive strain acting upon them from the 
load at J, will be more than large enough to resist the tensile 
strain upon them during the laying of the iirst part of the 
arch-stones below (3^ and c. 

415. — In the construction of arches, the voussoirs, or arch- 
stones, are so shaped that the joints between them are perpen- 
dicular to the curve of the arch, or to its tangent at the point 
at which the joint intersects the curve. In a circular arch, the 
joints tend toward the centre of the circle: in an elliptical 
arch, the joints may be found by the following process : 




304: 



AMERICAN HOUSE-CARPENTEE. 



416. — To find the direction of the joints for an elliptical 
arch, A joint being wanted at «, {Fig. 260,) draw lines from 
that point to tlie foci,/" and/*/ bisect the angle, /"(^y, with the 
line, al ; then a h will be the direction of the joint. 



/ l)rc 


f 


^lA 


e \ 


/ * 





Fig. 261. 



417. — To find the direction of the joimis for a parabolic arch 
A joint being wanted at a, {Fig. 261,) draw a e^ at right angles 
to the axis, eg ; make c g equal to c e^ and join a and g j draw 
a A, at right angles to ag j then a h will be the direction of 
the joint. The direction of the joint from h is found in the 
same manner. The lines, a g and hf are tangents to the curve 
at those points respectively ; and any number of joints in the 
curve may be obtained, by first ascertaining the tangents, and 
then drawing lines at right angles to them. 



JOINTS. 



Fig. 2C2. 

418. — Fig. 262 shows a simple and quite strong method of 
lengthening a tie-beam ; but the strength consists wholly in 
the bolts, and in the friction of the parts produced by screwing 
the pieces firmly together. Should the timber shrink to even 
a small degree, the strength would depend altogether on the 
bolts. It would be made much stronger by indenting the 
pieces together ; as at the upper edge of the tie-beam in Fig, 
263 ; or by placing keys in the joints, as at the lower edge in 



FKAMING. 305 

the same figure. This process, however, weakens the beam in 
proportion to the depth of the indents. 



r^ 



-CZI- 



Fig. 263. 

419. — Fig, 264 shows a method of scarfing, or splicing, a 
ne-beam without bolts. The keys are to be of well-seasoned, 

Fig. 264. 

iiard wood, and, if possible, very cross-grained. The addition 
of bolts would make this a very strong splice, or even white- 
oak pins would add materially to its strength. 



Fig. 265. 

■ 420. — Fig. 265 shows about as strong a splice, perhaps, as 
can well be made. It is to be recommended for its simplicity ; 
as, on account of there being no oblique joints in it, it can be 
readily and accurately executed. A complicated joint is the 
worst that can be adopted ; still, some have proposed joints 
that seem to have little else besides complication to recom 
mend them. 

421. — In proportioning the parts of these scarfs, the depths 
of all the indents taken together should be equal to one-third 
of the depth of the beam. In oak, ash or elm, the whole 
length of the scarf should be six times the depth, or thickness,, 
of the beam, when there are no bolts ; but, if bolts instead of 
indents are used, then three times the breadth ; and, when both 
methods are combined, twice the depth of the beam. The 



306 AilESICAN HOUSE-CAKPENTEE. 

lengtli of the scarf in pine and similar soft woods, depending 
wholly on indents, should be about 12 times the thickness, or 
depth, of the beam ; when depending wholly on bolts, 6 times 
the breadth ; and, when both methods are combined, 4 times 
the depth. 




422. — Sometimes beams have to be pieced that are required 
to resist cross strains — such as a girder, or the tie-beam of a 
roof when supporting the ceiling. In such beams, the fibres 
of the wood in the upper part are compressed ; and therefore 
a simple butt joint at that place, (as in Fig. 266,) is far prefer- 
able to any other. In such case, an oblique joint is the very 
worst. The under side of the beam being in a state of tension, 
it must be indented or bolted, or both ; and an iron plate un- 
der the heads of the bolts, gives a great addition of strength. 

Scarfing requires accuracy and care, as all the indents should 
bear equally ; otherwise, one being strained more than another, 
there would be a tendency to splinter off the parts. Hence 
the simplest form that will attain the object, is by far the best. 
In all beams that are compressed endwise, abutting joints, 
formed at right angles to the direction of their length, are at 
once the simplest and the best For a temporary purpose, Fig. 
262 would do very well ; it would be improved, however, by 
having a piece bolted on all four sides. Fig. 263, and indeed 
each of the others, since they have no oblique joints, would 
resist compression well. 

423. — In framing one beam into another for bearing pur- 
poses, such as a floor-beam into a trimmer, the best place to 
make the mortice in the trimmer is in the neutral line, {Arts. 
317, 318,) which is in the middle of its depth. Some have 
thought that, as the fibres of the upper edge are compressed, a 



FRAMING. 307 

mortice might be made tliere, and the tenon driveL in tight 
enough to make the parts as capable of resisting the compres- 
sion, as they would be without it ; and they have therefore 
concluded that plan to be the best. This could not be the case, 
even if the tenon would not shrink ; for a joint between two 
pieces cannot possibly be made to resist compression, so well 
as a solid piece without joints. The proper place, therefore, 
for the mortice, is at the middle of the depth of the beam ; but 
the best place for the tenon, in the floor-beam, is at its bottom 
edge. For the nearer this is placed to the upper edge, the 
greater is the liability for it to splinter off; if the joint is 



Fig. 261 

formed, therefore, as at Fig. 267, it will combine all the ad- 
vantages that can be obtained. Double tenons are objection- 
able, because the piece framed into is needlessly weakened, 
and the tenons are seldom so accurately made as to bear 
equally. For this reason, unless the tusk at a in the figure fits 
exactly, so as to bear equally with the tenon, it had better be 
omitted. And in sawing the shoulders, care should be taken 
not to saw into the tenon in the least, as it would wound the 
beam in the place least able to bear it. 

42i. — Thus it will be seen that framing weakens both pieces, 
more or less. It should, therefore, be avoided as much as pos- 
sible ; and where it is practicable one piece sliould rest ujpon 
the other, rather than be framed into it. This remark applies 
to the bearing of floor-beams on a girder, to the purlins and 
jack-rafters of a roof, &c. 

425. — In a framed truss for a roof, bridge, partition, &c., 
the joints should be so constructed as to direct the pressures 



308 



A3IEKICAN HOUSE-CAPwPENTEE. 



througli the axes of the several pieces, and also to avoid every 
tendency of the parts to slide. To attain this object, the abut- 




Fig. 268. 



Fig. 269. 



Fig. 270. 



ting surface on the end of a strut should be at right angles to 
the direction of the pressure ; as at the joint shown in Fig. 
268 for the foot of a rafter, (see Art. 277,) in Fig. 269 for the 
head of a rafter, and in Fig. 270 for the foot of a strut or 
brace. The joint at Fig. 268 is not cut completely across the 
tie-beam, but a narrow lip is left standing in the middle, and 
a corresponding indent is made in the rafter, to prevent the 
parts from separating sideways. The abutting surface should 
be made as large as the attainment of other necessary objects 
will admit. The iron strap is added to prevent the rafter slid- 
ing out, should the end of the tie-beam, by decay or otherwise, 
splinter off. In making the joint shown at Fig. 269, it should 
be left a little open at a, so as to bring the parts to a fair bear- 
ing at the settling of the truss, which must necessarily take 
place from the shrinking of the king-post and other parts. If 
the joint is made fair at first, when the truss settles it will cause 
it to open at the under side of the rafter, thus throwing the 
whole pressure upon the sharp edge at a. This will cause an 
indentation in the king-post, by which the truss will be made 
to settle further ; and this pressure not being in the axis of the 
rafter, it will be greatly increased, thereby rendering the rafter 
liable to split and break. 

426. — If the rafters and struts were made to abut end to 
end, as in Figs. 271, 272 and 273, and the king or queen post 
notched on in halves and bolted, the ill effects of shrinking 



FKAMING. 



309 



would be avoided. This metliod has been practised with suc- 
cess, iu some of the most celebrated bridges and roofs in Eu- 



^^ 



nSf 




<^-' 



Fig. 271. 



Fig. 272. 



/U 



Fig. 273. 



rope ; and, were its use adopted in this country, the unseemly 
sight of a hogged ridge would seldom be met with. A plate 
of cast iron between the abutting surfaces will equalize the 
pressure. 





Fig. 2T4. 



Fig. 275. 



427. — Fig. 274 is a proper joint for a collar-beam in a small 
roof: the principle shown here should characterize all tie- 
joints. The dovetail joint, although extensively practised iu 
the above and similar cases, is the very w^orst that can be em- 
ployed. The shrinking of the timber, if only to a small de- 
gree, permits the tie to withdraw — as is shown at Fig. 275. 
The dotted line shows the position of the tie after it has 
shrunk. 

428. — Locust and white-oak pins are great additions to the 
strength of a joint. In many cases they would supply the 
place of iron bolts ; and, on account of their small cost, they 
should be used in preference wherever the strength of iron is 



310 AMEKICAN HOUSE-CAEPENTEE. 

not requisite. In small framing, good cnt nails are of great 
service at the joints ; but they should not be trusted to bear 
any considerable pressure, as they are apt to be brittle. Iron 
straps are seldom necessary, as all the joinings in carpentry 
may be made without them. They can be used to advantage, 
however, at the foot of suspending-pieces, and for the rafter at 
the end of the tie-beam. In roofs for ordinaiy purposes, the 
iron straps for suspending-pieces may be as follows : "When the 
longest unsupported part of the tie-beam is 

10 feet, the strap may be 1 inch wide by -j^ thick. 



15 


a 


a 


a 


14 " 


U 


i 


20 


u 


u 


u 


2 " 


u 


i 



In fastening a strap, its hold on the suspending-piece vrill be 
much increased by turning its ends into the wood. Iron straps 
should be protected from rust ; for thin plates of iron decay 
very soon, especially when exposed to dampness. For this 
purpose, as soon as the strap is made, let it be heated to about 
a blue heat, and, while it is hot, pour over its entire surface 
raw linseed oil, or rub it with beeswax. Either of these will 
give it a coating which dampness will not penetrate. 



IRON GIRDEES. 




1 



Fig. 27CL Fig. 277. 

429. — Fig. 276 represents the front view, and Fig. 277 the 
cross section at middle, of a cast iron girder of proper form 
for sustaining a weight equally diffused over its length. The 
curve is that of a parabola : generally an arc of a circle is 



FRAMING. 311 

used, and is near enongli. Beams of this form are much used 
to sustain brick walls of buildings ; the brickwork resting upon 
the bottom flange, and laid, not arching, but horizontal. In 
the cross section, the bottom flange is made to contain in area 
four times as much as the top flange. The strength will be in 
proportion to the area of the bottom, flange, and to the height 
or depth. Hence, to obtain the greatest strength from a given 
amount of material, it is requisite to make the upright part, or 
the blade, rather thin ; yet, in order to prevent injurious strains 
in the casting while it is cooling, the parts should be nearly 
equal in thickness. The thickness of the three parts — blade, 
top flange and bottom flange, may be made in proportion as 5, 
.6 and 8. For a beam of this form, the weight equally dif- 
fused over it equals 

' ^=.9000^. (199.) 



The depth equals 

Iw , 

9000 ta ^ ' 

The area of the bottom flange equals 

<* = 90007^ ^^''-^ 

where w equals the weight in pounds equally diflfused over the 
length ; d, the depth, or height in inches of the cross section 
at middle ; <z, the area of the bottom flange in inches ; /, the 
length of the beam in feet, in the clear betw^een the bearings ; 
and ^, a decimal in proportion to unity as the safe weight is to 
the breaking weight. This is usually from 0-2 to 0*3, or from 
one-fifth to one-third, at discretion. 

430. — ^Beams of this form, laid in series, are much used in 
sustaining brick arches turned over vaults and other flre-proof 
rooms, forming a roof to the vault or room, and a floor above ; 
the arches springing from the flanges, one on either side of the 
beam, as in Fig. 278. 



312 



AMERICAN H0U6E-CAEPENTEK. 




ri?. 278. 



For this use the depth of cross section at middle equals 

(202.) 



9000 t a 



The area of the bottom flange equals 
cfP 



a z= 



(203.) 



9000 t d' 

where the symbols signify as before, and c equals the distance 
apart from centres in feet at which the beams are placed, and 
f the weight per superficial foot, in pounds, including the 
weight of the material of which the floor is constructed. 



Practical Rules and Examples. 

431. — For a single girder the dimensions may be found by 
the following rule, ((200) and (201) :) 

Biole LXYIII. — Divide the weight in pounds equally dif- 
fused over the length of the girder by a decimal in proportion 
to unity as the safe weight is to the breaking weight, multiply 
tlie quotient by the length in feet, and divide the product by 
9000. Then this quotient, divided by the depth of the beam 
at middle, will give the area of the bottom flange ; or, if di- 
vided by the area of the bottom flange, will give the depth — 
the area and depth both in inches. 

Example. — Let the weight equally difi*used over a girder 
equal 60000 pounds ; the decimal that is in proportion to unity 
as the safe weight is to be to the breaking weight, equal 0*3 ; 



FEAlSHNGi 313 

the length in the clear of the bearings equal 20 feet. Then 
60000 divided by 0-3 equals 200000, and this by 20 equals 
4000000 ; this divided by 9000 equals 4M|. Now if the depth 
is fixed, say at 20 inches, then 444^, divided by 20, equals 22|, 
equals the area of the bottom flange in inches. But if the 
area is given, say 24 inches, then to find the depth, divide 
444f by 24, and the quotient, 18-5, equals the depth in inches ; 
and such a girder may be made with a bottom flange of 2 by 
12 inches, top flange, (equal to \ of bottom flange,) l-J by 4 
inches, and the blade 1^ inches thick. 

432. — For a series of girders or iron beams, the dimensions 
may be found by the following rule : (202) and (203). 

Rule LXIX. — Divide the weight per superficial foot, in 
pounds, by a decimal in proportion to unity as the safe weight 
is to the breaking weight, and multiply the quotient by the 
square of the length of the beams and by the distance apart at 
which the beams are placed from centres, both in feet, and di- 
vide the product by 9000. Then this quotient, divided by the 
depth of the beams at middle, will give the area of the bottom 
flange ; or, if divided by the area of the bottom flange, will 
give the depth of the beam — the depth and area both in inches. 

Examjple. — Let the weight per superficial foot resting upon 
an arched floor be 200 pounds, and the weight of the arches, 
concrete, &c., equal 100 pounds, total 300 pounds per superfl- 
cial foot. Let the proportion of the breaking weight to be 
trusted on the beams equal 0*3, the length of the beams in the 
clear of the bearings equal 12 feet, and the distance apart from 
centres at which they are placed equal 4 feet. Then 300 di- 
vided by 0-3 equals 1000 ; this multiplied by 144 (the square 
of 12), equals 144000, and this by 4, equals 576000 ; this di- 
vided by 9000, equals 64. Now if the depth is fixed, and at 
8 inches, then 64 divided by 8 equals 8, equals the area of the 
bottom flange. But if the area of the bottom flange is fixed, 
and at 6 inches, then 64, divided by 6, equals lOf, the depth 

40 



314 



AMEEICAIs HOUSE-CAKPENTEE. 



required. Such a beam may be made with the bottom flange 
1 by 6 inches, the top flange, (equal to one-quarter of the bot- 
tom flange,) f by 2 inches, and the blade f inch thick. 

433. — The kind of girder shown at Fig. 280, (a cast iron 
arch with a wrought iron tie rod,) is extensively used as a sup- 
port upon which to build brick walls where the space below is 
required to be free. The objections to its use are, the dispro- 
portion between the material and the strains, and the enhanced 
cost over the cast iron girder formed as in Figs. 276 and 27T. 
The material in the cast arch, (Fig. 280,) is greatly in excess 
over the amount needed to resist effectually the compressive 
strains induced by the load through the axis of the arch, while 
the wrought metal in the tie is usually much less than is re- 
quired to resist the horizontal thrust of the arcli ; absolute fail- 
ure being prevented, partly by the weight of the walls resting 
on the haunches, and partly by the presence of adjoining 
buildings, their walls acting as buttresses to the arcli. Some 
instances have occuiTed where the tie has parted. 

Wh^re this arched girder is used it is customary to lay the 
first courses of brick in the form of an arch. This brick arch 
of itself is quite sufficient to sustain the compressive strain, 
and, were there proper resistance to the horizontal thrust pro- 
vided, the brick arch would entirely supersede the necessity 
for the girder. Indeed, the instances are not rare where con- 
structions of this nature have proved quite satisfactory, the 
horizontal thrust of the arch being sustained by a tie rod 
secured to a pair of cast iron heel plates, as in Fig. 279. The 




Fig. 2T9. 



FEAMTNG. 316 

brick arcli, in tliis case, being built upon a wooden centre, 
which was afterwards removed. 

The diameter of the rod required for an arch of this kind is 
equal to 

^=^mh ^'"'-^ 

where w equals the weight in pounds equally diffused over the 
arch ; s, the length of the rod, clear of the heel plates, in feet ; 
and A, the height at the middle, or rise, of the arc, in inches ; 
I), the diameter, being also in inches. 

When the diameter found by formula (204) is impracticably 
large, this difficulty may be overcome by dividing the metal 
into two rods. In the bow-string girder, {Mg. 280,) two rods 
cannot be used with advantage, because of the difficulty in 
adjusting their lengths so as to ensure to each an equal amount 
of the strain. But in the case of the brick arch, the two heel 
plates being disconnected, any discrepancy of length in the 
rods is adjusted simply by the pressure of tlie arch acting on 
the plates. When there are to be two rods, the diameter of 
each rod equals 



Practical Rule and Example. 

434. — ^To obtain the diameter of wrought iron tie-rods foi 
heel plates, as in Fig. 279, proceed by this rule. 

Hule LXX. — Multiply the weight in pounds equally distri- 
buted over the arch by the length of the tie-rod in feet, cleai 
of the heel plates, and divide the product by the height of the 
arc in inches, (that is, the height at the middle, from the axis 
of the tie-rod to the centre of the depth of the brick arch,) then, 
if there is to be but one tie- rod, divide the quotient by 3000 ; 



316 



AMEKICAN H'OUSE-CAKPENTER. 



but if two, then divide by 6000, and the square root of the 
quotient, in either case, will be the required diameter. 

Exarrvple. — The weight to be supported on a brick arch, 
equally distributed, is 24000 pounds ; the length of the tie-rod, 
clear of the heel plates, is 10 feet ; and the height, or rise, of 
the arc is 10 inches. Now by the rule, 24000 X 10 = 240000. 
This divided by 10, equals 24000. Upon the presumption that 
one tie-rod only will be needed, divide by 3000, and the quo- 
tient is 8, the square root of which is 2*82 inches. This is 
rather large, therefore there had better be two rods. In this 
case the quotient, 24000, divided by 6000, equals 4, the square 
root of which is 2, the diameter required. The arch should, 
therefore, have two rods of 2 inches diameter. Two rods are 
preferable to one. The iron is stronger per inch in small rods 
than in large ones, and the rules require no more metal in the 
two rods than in the one. 




Fig. 280. 



435. — The Bow-si/ring Girder^ as per Fig. 280, has little to 
recommend it, (see Art. 433,) yet because it has by some been 
much used, it is well to show the rules that govern its strength, 
if only for the benefit of those who are willing to be governed 
by reason rather than precedent. To resist the horizontal 
thrust of the cast arch, the diameter of the rod must equal 
(204) 

~V3000A' 
But the cast iron arch has a certa'n amount of strength to re- 



FRAMING. 6U 

sist cross strains : this strengtli must be considered. Upon the 
presumption that the cross section of the cast arch at the mid- 
dle is of the most favorable form, as in I^ig. 277, or at least 
that it have a bottom flange, (although the most of those cast 
are without it), the strength of the cast arch to resist cross 
strains is shown by formula (199), when. I, its length, is changed 
to s, its span. The weight in pounds equally diffused over the 
arch will then equal 

9000 t a d 

w = . 

s 

This is the weight borne by the cast arch acting simply as a 
beam. Deducting this weight from the whole weight, the re- 
mainder is the weight to be sustained by the rod. Calling the 
whole weight w, then 

9000 tad ws- 9000 tad==W 
w — = 

s s 

Therefore, fiom (204), the diameter equals 



"'^i 



3000 A 

// «)s-9000ga<g \ 



^ S 
3000 A ^ 



Iws^ 9000 t a d 



(206.) 



3000 h 

where D equals the diameter of the rod in inches ; w^ the 
weight in pounds equally diffused over the arch ; 5, the span 
of the arch in feet ; A, the rise or height of the arc at middle, 
in inches ; J, the height or depth of the cross section of the 
cast arch in inches ; «, the area of the bottom flange of the 
cross section of the cast arch in inches ; and ^, a decimal in 
proportion to unity as the safe weight is to be to the breaking 
weight. 

The rule in words at length, is 

Rule LXXI. — ^Multiply the decimal in proportion to unity 



318 AMERICAN IIOUSE-CARPENTEK. 

as the safe weight is to be to the breaking weight, by (♦000 
times the depth of the cross section of the cast arch at middle, 
and by the area of the bottom flange of said section, both in 
inches, and deduct the product from the weight in pounds 
equally diffused over the arch multiplied by the span in feet, 
and divide the remainder by 3000 times the height of the arc 
in inches, measured from the axis of the tie-rod to the centre 
of the depth of the cast arch at middle, and the square root 
of the quotient will be the diameter of the rod in inches. 

Example. — ^The rear wall of a building is of brick, and is 40 
feet high, and 21 feet wide in the clear between the piers of 
the story below. Allowing for the voids for windows, this 
wall will weigh about 63000 pounds ; and it is proposed to 
support it by a bow-string girder, of which the cross section at 
middle of the cast arch is 8 inches deep, and has a bottom 
flange containing 12 inches area. The rise of the curve or arc 
is 24 inches. What must be the diameter of the rod, the por- 
tion of the breaking weight of the cast arch, considered safe 
to trust, being three-tenths or 0*3? By the rule, 0*3 x 9000 
X 8 X 12.= 259200; then 63000 x 21 - 259200 = 1063800. 
This remainder divided by (3000 X 24 =) 72000, the quotient 
equals 14*775 ; the square root of which, 3*84, or nearly 3| 
inches, is the required diameter. 

This size, though impracticably large, is as small as a due 
regard for safety will permit ; yet it is not unusual to find the 
rods in girders intended for as heavy a load as in this exam- 
ple, only 2i and 2^ inches ! Were it possible to attach the 
rod so as not to injure its strengtli in the process of shrinking 
it in — putting it to its place hot, and depending on the con- 
traction of the metal in cooling to bring it to a proper bearing 
— and were it possible to have the bearings so true as to induce 
the strain through the axis of the rod, and not along its side^ 
{Art. 308,) then a less diameter than that given by the rule 
would suffice. But while these contingencies remain, the rule 



FRAMING. 319 

cannot safely be reduced, for, in the rule, the value of T, for 
wrought iron, (Table III., Art. 308,) is taken at nearly 6000 
pounds, a point rather high in consideration of the size, of the 
rod and the injuries, before stated, to which it is subjected. 
In cases where a girder wholly of cast iron {Mg. 276) is not 
preferred, it were better to build a brick arch resting on heel 
plates, {Mg. 279,) in which the metal required to resist the 
thrust ma}^ be divided into two rods instead of one, thus render- 
ing the size more practical, and at the same time avoiding the 
injuries to which rods in arch girders are subjected. The heel- 
plate arch is also to be preferred to the cast arch on the score 
of economy ; inasmuch as the brick which is substituted for 
the east arch will cost less than iron. For example, suppose 
the cross section of the iron arch to be thus : the blade or up- 
right part 8 by 1-J- inches, the top flange 12 by IJ inches, and 
the bottom flange 6 by If inches. At these dimensions, the 
a^ea of the cross section will equal 12 + 15 + lOj- = 37-| 
inches. A bar of cast iron, one foot long and one inch square, 
will weigh 3-2 pounds; therefore, 37i X 3-2 = 120 pounds, 
equals the weight of the cast arch per lineal foot. The price 
of castings per pound, as also the price of brickwork per 
cubic foot, of course will depend upon the localit}^ and the 
state of the market at the time, but for a comparison they may 
be stated, the one at three and a half cents per pound, and the 
other at thirty cents per cubic foot. At these prices the cast 
arch will cost 120 x 3^ = $4 20 per lineal foot; while the 
brick arch — 12 inches high and 12 inches thick — will cost 30 
cents per lineal foot. The diff'erence is $3 90. This amount 
is not all to be credited to the account of the brick arch. 
I'roper allowance is to be made for the cost of the heel plates, 
and of the wooden centre ; also for the cost of a small addi- 
tion to the size of the tie rods, which is required to sustain the 
strain otherwise borne by the cast arch in its resistance to a 
cross strain {Art. 435). Deducting the cost of these items, 



320 AMERICAN HOUSE-CAErENTEE. 

the difference in favor of the brick arch will be about $3 per 
foot. This, on a girder 25 feet long, amounts to $75. The 
difference in all cases will not equal this, but will be sufficiently 
great to be worth saving. 



SECTION v.— DOORS, WINDOWS, <kc. 



DOORS. 

436. — Among the several architectural arrangements of an edi- 
fice, the door is by no means the least in importance ; and, if pro- 
perly constructed, it is not only an article of use, but also of or- 
nament, adding materially to the regularity and elegance of the 
apartments. The dimensions and style of finish of a door, should 
be in accordance with the size and style of the building, or the 
apartment for which it is designed. As regards the utility of 
doors, the principal door to a public building should be of suffi- 
cient width to admit of a free passage for a crowd of people ; 
while that of a private apartment will be wide enough, if it per- 
mit one person to pass without being incommoded. Experience 
has determined that the least width allowable for this is 2 feet 8 
inches ; although doors leading to inferior and unimportant rooms 
may, if circumstances require it, be as narrow as 2 feet 6 inches ; 
and doors for closets, where an entrance is seldom required, may 
be but 2 feet wide. The width of the principal door to a public 
building may be from 6 to 12 feet, according to the size of the 
building ; and the width of doors for a dwelling may be from 2 
feet 8 inches, to 3 feet 6 inches. If the importance of an apart- 
ment in a dwelling be such as to require a door of greater width 

41 



322 AMERICAN HOUSE-CARPENTER. 

than 3 feet 6 incheSj the opening should be closed with two 
doors, or a door in two folds ; generally, in such cases, where the 
opening is from 5 to 8 feet, folding or sliding doors are adopted. 
As to the height of a door, it should in no case be less than about 
6 feet 3 inches ; and generally not less than 6 feet 8 inches. 

437. — The proportion between the width and height of single 
doors, for a dwelling, should be as 2 is to 5 ; and, for entrance- 
doors to public buildings, as 1 is to 2. If the width is given and 
the height required of a door for a dwelling, multiply the width 
by 5, and divide the product by 2 ; but, if the height is given and 
the width required, divide by 5, and multiply by 2. Where two 
or more doors of different widths show in the same room, it is 
well to proportion the dimensions of the more important by the 
above rule, and make the narrower doors of the same height as 
the wider ones ; as all the doors in a suit of apartments, except 
the folding or sliding doors, have the best appearance when of 
one height. The proportions for folding or sliding doors should 
be such that the width may be equal to J of the height ; yet this 
rule needs some qualification : for, if the width of the opening 
be greater than one-half the width of the room, there will not be 
a sufficient space left for opening the doors ; also, the height 
should be about one-tenth greater than that of the adjacent single 
doors. 

438. — Where doors have but two panels in width, let the stiles 
and muntins be each 4 of the width ; or, whatever number of 
panels there may be, let the united widths of the stiles and the 
muntins, or the whole width of the solid, be equal to | of the width 
of the door. Thus : in a door, 35 inches wide, containing two 
panels in width, the stiles should be 5 inches wide ; and in a door, 
3 feet 6 inches wide, the stiles should be 6 inches. If a door, 3 
feet '6 inches wide, is to have 3 panels in width, the stiles and 
muntins should be each 4^ inches wide, each panel being 8 inches. 
The bottom rail and the lock rail ought to be each equal in 
width to y of the height of the door ; and the top rail, and all 



DOORS, WINDOWS, &C. 



323 



others, of the same width as the stiles. The moulding on the 
panel should be equal in width to i of the width of the stile. 




LJ^ 



Fig. 281. 



439. — Fig. 281 shows aji approved method of trimming doors : 
a is the door stud ; 6, the lath and plaster ; c, the ground ; d, the 
jamb ; e, the stop ; /and g, architrave casings ; and A, the door 
stile. It is customar}^ in ordinary work to form the stop for the 
door by rebating the jamb. But, when the door is thick and 
heavy, a better plan is to nail on a piece as at e in the figure. 
This piece can be fitted to the door, and put on after the door is 
hung ; so, should the door be a trifle winding, this will correct 
the evil, and the door be made to shut solid. 

440. — Fig. 282 is an elevation of a door and trimmings suita- 
ble for the best rooms of a dwelling. (For trimmings generally, 
see Sect. III.) The number of panels into which a door should 
be divided, is adjusted at pleasure ; yet the present style of finish- 
ing requires, that the number be as small as a proper regard for 
strength will admit. In some of our best dwellings, doors have 
been made having only two upright panels. A few years expe- 
ience, however, has proved that the omission of the lock rail 
is at the expense of the strength and durability of the door ; a 
four-panel door, therefore, is the best that can be made. 

441. — The doors of a dwelling should all be hung so as to open 
into the principal rooms ; and, in general, no door should be hung 
to open into the hall, or passage. As to the proper edge of the 
door on which to affix the hinges, no general rule can be assigned 



324: 



AMERICAN HOUSE-CAEPENTER. 




Fig. 288. 



WINDOWS. 



44:2. — A. window should be of such dimensions, and in such 
a position, as to admit a sufficiency of light to that part of the 
apartment for which it is designed. No definite rule for the size 



DOORS, WINDOWS, &.C. o2.) 

can well be given, that will answer in all cases ; yet, as an ap- 
proximation, the following has been used for general purposes. 
Multiply together the length and the breadth in feet of the apart- 
ment to be lighted, and the product by the height in feet ; then 
the square-root of this product will show the required number of 
square feet of glass. 

443. — To ascertain the dimensions of window frames, add 4|^ 
inches to the width of the glass for their width, and 6^ inches to 
the height of the glass for their height. These give the dimen- 
sions, in the clear, of ordinary frames for 12-light windows ; the 
height being taken at the inside edge of the sill. In a brick wall, 
the width of the opening is 8 inches more than the width of the 
s^lass — 4§ for the stiles of the sash, and 3| for hanging stiles — ■ 
and the height between the stone sill and lintel is about 10 § inches 
more than the height of the glass, it being varied according to the 
thickness of the sill of the frame. 

444. — In hanging inside shutters to fold into boxes, it is ne- 
cessary to have the box shutter about one inch wider than the 
flap, in order that the flap may not interfere when both are folded 
mto the box. The usual margin shown between the face of the 
shutter when folded into the box and the quirk of the stop bead, 
or edge of the casing, is half an inch ; and, in the usual method 
sof letting the whole of the thickness of the butt hinge into the 
fcdge of the box shutter, it is necessary to make allowance for the 
throw of the hinge. This may, in general, be estimated at ? of 
an inch at each hinging ; which being added to the margin, the 
entire width of the shutters will be 1^ inches more than the width 
of the frame in the clear. Then, to ascertain the width of the 
box shutter, add 1 5 inches to the width of the frame in the clear, 
between the pulley stiles ; divide this product by 4, and add 
half an inch to the quotient ; and the last product will be the re- 
quired width. For example, suppose the window to have 3 
lights in width, 11 inches each. Then, 3 times 11 is 33, and 4^ 
added for the wood of the sash, gives 37^ STj and 1;)^ is 39 



326 AMERICAN HOUSE CARPENTER. 

and 39; divided by 4, gives 9 J ; to which add half an inch, and 
the result will be 10^ inches, the width required for the box shutter. 
4:4:5. — In disposing and proportioning windows for the walls of 
a building, the rules of architectural taste require that they be af 
different heights in different stories, but of the same width. The 
windows of the upper stories should all range perpendicularly 
over those of the first, or principal, story ; and they should be 
disposed so as to exhibit a balance of parts throughout the front 
of the building. To aid in this, it is always proper to pl.-ice the 
front door in the middle of the front of the building ; and, where 
the size of the house will admit of it, this plan should be adopted. 
(See the latter part of A?'t. 224.) The proportion that the height 
should bear to the width, may be, in accordance with general 
usage, as follows : 

The height of basement windows, 1^ of the width. 
" " principal-story " 2| " 

" " second-story " 1| " 

" " third-story " 1^ ' 

" " fourth-story " 1,^ " 

" " attic-story " the same as the width. 

But, in determining the height of the windows for the several 
stories, it is necessary to take into consideration the height of the 
story in which the window is to be placed. For, in addition to 
the height from the floor, which is generally required to be from 
28 to 30 inches, room is wanted above the head of the window 
for the window-trimming and the cornice of the room, besides 
some respectable space which there ought to be between these. 

446. — Doors and windows are usually square- headed, or termi- 
nate in a horizontal line at top. These require no special direc- 
tions for their trimmings. But circular-headed doors and win- 
dows are more difficult of execution, and require some attention. 
If the jambs of a door or window be placed at right angles to the 
face of the wall, the edges of the soffit, or surface of the head, 
would be straight, and its length be found by getting the 



DOORS, WINDOWS, «fcC. 



32< 



6tretch-out of the circle, {Art. 92;) but, when the jaiibs are 
placed obliquely to the face of the wall, occasioned by the de- 
mand for light in an oblique direction, the form of the soffit 
will be obtained by the following article : and, when the face 
of the wall is circular, as in the succeeding one. 

/ . 





7-= 


T=^ 










_U 


c 






= 


N 


\ 






5-^ 


^"^ 


5^ 




' — ^ 


— 


~ ■■ 


3 


r- 


^6 




a' — 


rn 


t 


' — -J 






^=4 


^ 






— -1 


^\ 




-—J 


-=l 


=t 


w 




T\ \ \) 



Fig. 288. 



447. — To find the form of the soffit for circular window 
heads, when the light is received in an oblique direction. Let 
abed, {Fig. 283,) be the ground-plan of a given window, and ef 
a, a vertical section taken at right angles to the face of the jambs. 
From a, through e, draw a g, at right angles to a 6 ; obtain the 
stretch-out of ef a, and make e g equal to it ; divide e g and e 
f a, each into a like number of equal parts, and drop perpen- 
diculars from the points of division in each ; from the points of 
intersection, 1, 2, 3, &c., in the line, a d, draw horizontal lines to 
meet corresponding perpendiculars from eg; then those points 
of intersection will give the curve line, d g, which will be the 
one required for the edge of the soffit. The other edge, c h, is 
found in the same manner. 

448. — To find the form of the soffit for circular toindov^- 
heads, when the face of the wall is curved. Let abed, {Fig. 
284,) be the ground-plan of a given window, and efa,^ vertical 
section of the head taken at right angles to the face of the jambs. 



328 



AMERICAN HOUSE-CARPENTER. 




Fig. 284. 

Proceed as in the foregoing article to obtain the line, d g; then 
that will be the curve required for the edge of the soffit,- the 
other edge being found in the same manner. 

If the given vertical section be taken in a line with the face of 
the wall, instead of at right angles to the face of the jambs, place 
it upon the line, c h, {Fig. 283 ;) and, having drawn ordinates at 
right angles to c b, transfer them to ef a ; in this way, a section 
at right angles to the jambs can be obtained. 



SECTION VL— STAIRS. 



4tt9. — The STAIRS is that mechanical arrangement in a build- 
ing by which access is obtained from one story to another. Their 
position, form and finish, when determined with discriminating 
taste, add greatly to the comfort and elegance of a structure. As 
regards their position, the first object should be to have them near 
the middle of the building, in order that an <isqually easy access 
may be obtained from all the rooms and passages. Next in im- 
portance is light ; to obtain which they would seem to be best 
situated near an outer wall, in which ivindows might be construc- 
ted for the purpose ; yet a sky-light, or opening in the roof, would 
not only provide light, and so secure a central position for the 
stairs, but may be made, also, to assist materially as an ornament 
to the building, and, what is of more importance, afford an op- 
portunity for better ventilation. 

450. — It would seem that the length of the raking side of the 
intch-board, or the distance from the top of one riser to the top ot 
the next, should be about tlie same in all cases ; for, whether stairs 
be intended for large buildings or for small, for public or for pri- 
vate, the accommodation of men of the same stature is to be con- 
sulted in every instance. But it is evident that, with the same 
effort, a longer step can be taken on level than on rising ground ; 

42 



330 



AMERICAN HOUSE-CARPENTER. 



and that, although the tread and rise cannot be proportioned 
merely in accordance with the style and importance of the build- 
ing, yet this may be done according to the angle at which the 
fjght rises. If it is required to ascend gradually and easy, the 
length from the top of one rise to that of another, or the hypothe 
nuse of the pitch-board, may be long ; but, if the flight is steep 
the length must be shorter. Upon this data the folio wiug pioblenr 
is constructed. 




451. — To proportion the rise and tread to one another. 
Make the line, a 6, {Fig. 285,) equal to 24 inches ; from 6, ereci 
h c, at right angles to a b, and make b c equal to 12 inches ; join a 
and c, and the triangle, a b c, will form a scale upon which to 
graduate the sides of the pitch-board. For example, suppose a 
very easy stairs is required, and the tread is fixed at 14 inches. 
Place it from b to/, and from/; draw/^, at right angles to a b ; 
then the length of f g will be found to be 5 inches, which is a 
proper rise for 14 inches tread, and the angle, f b g, will show 
the degree of inclination at which the flight will ascend. But, in 
a majority of instances, the height of a story is fixed, while the 
length of tread, or the space that the stairs occupy on the lower 
floor, is optional. The height of a story being determined, the 
height of each rise will of course depend upon the number, into 
which the whole height is divided ; the angle of ascent being more 
easy if the number be great, than if it be smaller. By dividing 



STAIRS. 331 

the whole height oi' a story into a certain number of rises, sup- 
pose the length of each is found to be 6 inches. Place this length 
from b to A, and draw h i, parallel to a b ; then h i, or bj will be 
the proper tread for that rise, and 7 b i will show the angle of as- 
cent. On the other hand, if the angle of ascent be given, as a 
b I, {b I being 10^ inches, the proper length of run for a step- 
ladder,) drop the perpendicular, I k, from I to k ; then I kb will 
be the proper proportion for the sides of a pitch-board for that 
run. 

452. — The angle of ascent will vary according to circum- 
stances. The following treads will determine about the right in- 
clination for the different classes of buildings specified. 

In public edifices, tread about 14 inches. 

In first-class dwellings " 12-| " 

In second-class " •" 11 ^* 

In third-class " and cottages " 9 " 

Step-ladders to ascend to scuttles, (fcc, should have from 10 to 
11 inches run on the rake of the string. (See notes at Art. 103.) 
453. — The length of the steps is regulated according to the ex- 
tent and importance of the building in which they are placed, 
varying from 3 to 12 feet, and sometimes longer. Where two per- 
sons are expected to pass each other conveniently, the shortest 
length that will admit of it is 3 feet ; still, in crowded cities where 
land is so valuable, the space allowed for passages being very 
small, they are frequently executed at 2\ feet. 

454. — To find the dimensions of the pitch-board. The first 
thing in commencing to build a stairs, is to make the pi^cA-board ; 
this is done in the following manner. Obtain very accurately, in 
feet and inches, the perpendicular height of the story in which 
the stairs are to be placed. This must be taken from the top ol 
the floor in the lower story to the top of the floor in the upper 
story. Then, to obtain the number of rises, the height in inches 
thus obtained must be divided by 5, 6, 7, 8, or 9, according to the 
quality and style of the building in which the stairs are to be 



332 AMERICAN HOUSE-CARPENTER. 

built. For instance, suppose the building to be a fiist-class 
dwelling, and the height ascertained is 13 feet 4 inches, or 160 
inches. The proper rise for a stairs in a house of this class is 
about 6 inches. Then, 160 divided by 6, gives 26t i«ches This 
being nearer 27 than 26, the number of risers, should be 27. 
Then divide the height, 160 inches, by 27, and the quotient will 
give the height of one rise. On performing this operation, the 
cfuotient will be found to be 5 inches, | and ~ of an inch. 

Then, if the space for the extension of the stairs is not limited, 
the tread can be found as at Art. 451. But, if the contrary is the 
case, the whole distance given for the treads must be divided by 
the number of treads required. On account of the upper flooi 
forming a step for the last riser, the number of treads is always 
one less than the number of risers. Having obtained this 
rise and tread, the pitch-board may be made in the follow- 
ing manner. Upon a piece of well-seasoned board about f of an 
inch thick, having one edge jointed straight and square, lay the 
corner of a carpenters'-square, as shown at Fig. 286. Make a b 




Fig 286. 

equal to the rise, and b c equal to the tread ; mark along those 
edges with a knife, and cut it out by the marks, making the edges 
perfectly square. The grain of the wood must run in the direction 
indicated in the figure, because, if it shrinks a trifle, the rise and 
the tread will be equally affected by it. When a pitch-board is 
first made, the dimensions of the rise and tread should be pre- 
served in figures, in order that, should the first shrink, a second 
could be made. 
455. — To lay out the string. The space required for timbei 



STAIRS. 



333 




Fig. 287. 



and plastering under the steps, is about 5 inches for ordinary 
stairs ; set a gauge, therefore, at 5 inches, and run it on the lower 
edge of the plank, as a b, {Fig. 287.) Commencing at one end, 
lay the longest side of the pitch-board against the gauge-mark, a 
b, as at c, and draw by the edges the lines for the first rise and 
tread ; then place it successively as at d, e and /, until the re- 
quired number of risers shall be laid down. 



w 



E7 



ni 



Fig. 288. 



456. — Fig. 288 represents a section of a step and riser, joined 
after the most approved method. In this, a represents the end of 
a block about 2 inches long, two of which are glued in the corner 
in the length of the step. The cove at b is planed up square, 
glued in, and stuck after the glue is set. 



PLATFORM STAIRS. 

457. — A platform stairs ascends from one story to another in 
two or more flights, having platforms between for resting and 
to en ange their direction. This kind of stairs is the most easily 
constructed, and is therefore the most common. The cylin- 



su 



AMERICAN HOUSE-CARPENTER 




Fig. 289. 



der is generally of small diameter, in most cases about 6 inches. 
It may be worked out of one solid piece, but a better way is lo 
glue together three pieces, as in Fig, 289 ; in which the pieces j 
a, b and c, compose the cylinder, and d and e represent parts of 
the strings. The strings, after being glued to the cylinder, are 
secured with screws. The joining at o and o is the most proper 
for that kind of joint. 

4:58. — To obtain the form of the lower edge of the cylinder. 
Find the stretch-out, d e, {Fig. 290,) of the face of the cylinder 
a b c, according to Art. 92 ; from d and e, draw d f and e g, at 
right angles to d e ; draw h g^ parallel to d e, and make hf and 
g ij each equal to one rise; from i and/, draw ij andfk, paral- 
lel to h g ; place the tread of the pitch-board at these last lines, 
and draw by the lower edge the lines, k h and i I ; parallel to 
these, draw m n and o />, at the requisite distance for the dimen- 
sions of the string ; from 5, the centre of the plan, draw v«? q, 
parallel to df; divide h qand q g, each into 2 equal parts, as at 
V and to; from v and w?, draw v n and lo o, parallel tofd; join n 
and 0, cutting q s in r ; then the angles, u n r and r o t. being 
eased off according to Art. 89, will give the proper curve for the 
bottom edge of the cylinder. A centre may be found upon which 
to describe these curves thus : from u, draw u x, at right angles 
to m n ; Irom r, draw r .v, at right angles to n o ; then a: will be 
the centre for the curve, u r. The centre for the ;mfve. r t, is 
foimd in the same manner. 



STAIRS. 



8;^5 




Fig. 290. 



459. — To find the position for the balusters. Place the 
centre of the first baluster, ih. Fig. 291,) I its diameter from the 
face of the riser, c c?, and i its diameter from the end of the step, 
e d ; and place the centre of the other baluster, a, half the tread 
from the centre of the first. The centre of the rail must be placed 
over the centre of the balusters. Their usual length is 2 feet 
5 inches, and 2 feet 9 inches, for the short and the long balusters 
respectively. 



.-&^^i 



Fig. 291. 



336 



AMERICAN HOUSE-CARPENTER. 




460. — To find the face-mould for a round hand-rail to plat- 
form stairs. Case 1. — When the cylinder is small. In Fig. 

292, j and e represent a vertical section of the last two steps of the 
first flight, and d and i the first two steps of the second flight, of 
a platform stairs, the line, e f being the platform ; and a b c is 
the plan of a line passing through the centre of the rail around 
the cylinder. Through i and d, draw i k, and through J and e, 
draw 7 k ; from k, draw k I, parallel to f e ; from 6, draw b m, 
parallel tog d; from /, draw I r, parallel to k j ; from n, draw 7i 
t, at right angles toj k : on the Hue, o b. make o t equal to n t ; 
join c and t : on the line, 7 c, {Fi^. 293 ) make e c equal to en at 
Fig. 292 ; from c, draw c t, at right angles toj c, and make c t 



STAIRS. 337 

i I 




equal to c ^ at Fig. 292 ; through ^, drawp /, parallel to^' c, and 
make 1 1 equal to ^ Z at Fig. 292 ; join I and c, and complete the 
parallelogram, eels; find the points, o, o, o, according to Art. 
118 ; upon e, o, o, o, audi, successively, with a radius equal to 
half the width of the rail, describe the circles shown in the figure ; 
then a curve traced on both sides of these circles and just touch- 
ing them, will give the proper form for the mould. The joint at 
I is drawn at right angles to c I. 

4:61. — Elucidation of the foregoing method. This excellent 
plan for obtaining the face-moulds for the hand-rail of a platform 
stairs, has never before been published. It was communicated to 
me by an eminent stair-builder of this city : and having seen 
rails put up from it, I am enabled to give it my unqualified re- 
commendation. In order to have it fully understood, I have in- 
troduced Fig. 294 ; in which the cylinder, for this purpose, is 
made rectangular instead of circular. The figure gives a per- 
spective view of a part of the upper and of the lower flights, and 
a part of the platform about the cylinder. The heavy lines, i m, 
m c and cj, show the direction of the rail, and are supposed to 
pass through the centre of it. When the rake of the second 
flight is the same as that of the first, whicL is here and is gene- 
rally the case, the face-mould for the lower twist will, when rti- 
versed, do for the upper flight : that part of the rail, therefore, 
which passes from e to c and from c to I, is all that will need ex- 
planation. 

Suppose, then, that the parallelogram, e a o c, represent a pleine 
lying perpendicularly over e abf being inclined in the direction, 
e c, and level in the direction, c o ; suppose this plane, e a o Cj 

43 



338 



AMERICAN HOUSE-CARPENTER. 




Fig. 294. 



be revolved on e c as an axis, in the manner indicated by the arcs, 
n and a ar, until it coincides with the plane, e r t c ; the line, a 
0, will then be represented by the line, a; n ; then add the paral- 
lelogram, xrt n, and the triangle, ctl, deducting the triangle, ers; 
and the edges of the plane, e s I c, inclined in the direction, ec, and 
also in the direction, c I, will lie perpendicularly over the plane, e 
a bf. From this we gather that the line, c o, being at right angles to 



STAIRS. 



339 



e c, must, in order to reach the point, Z, be lengthened the distance, 
n t, and the right angle, e c ?, be made obtuse by the addition to 
it of the angle, t c I. By reference to Fig. 292, it will be seen 
that this lengthening is performed by forming the right-angled 
triangle, c o t^ corresponding to the triangle, co t/va Fig. 294. 
The line, c #, is then transferred to Fig. 293, and placed at right 
angles to e c; this angle, e c tj being increased by adding the an- 
gle, t cl, corresponding to ^ c Z, Fig. 294, the point, Z, is reached, 
and the proper position and length of the lines, e c and c I ob- 
tained. To obtain the face-mould for a rail over a cylindrical 
well-hole, the same process is necessary to be followed until the 
the length and position of these lines are found ; then, by forming 
the parallelogram, eels, and describing a quarter of an ellipse 
therein, the proper form will be given. 




Fig 295. 



462.— Case 2 -^ When the cylinder is large. Fig. 295 re- 



uo 



AMERICAN HOUSE-CARPENTER. 



presents a plan and a vertical section of a line passing through the 
centre of the rail as before. From b, draw b k, parallel to cd; ex- 
tend the lines, i d and J e, until they meet kbink and/; from n, 
draw n I, parallel to ob ; through Z, draw 1 1, parallel tojk, from 
k^ draw k t, at right angles Xoj k ; on the line, o 6, make o t equal 
to k t. Make e c, [Fig. 296,) equal to e A: at Fig. 295 ; from c, 




Fig. 296. 

draw c t^ at right angles to e c, and equal to c if at Fig. 295 ; from 
^, draw t p, parallel to c e, and make 1 1 equal to ^ / at Fig. 295 ; 
complete the parallelogram, eels, and find the points, o, o, o, as 
before ; then describe the circles and complete the mould as in 
Fig. 293 The difference between this and Case 1 is, that the 
line, c t, instead of being raised and thrown out, is lowered and 
drawn in. (See note at page 381.) 




Fig. 297. c 

463. — Case 3. — Where the rake meets the level. In Fig 



STAIRS. 



341 



297, ab CIS the plan of a line passing through the centre ct tne 
rail around the cylinder as before, and 7 and e is a vertical section 
of two steps starting from the floor, h g. Bisect ehmd^ and 
through rf, draw c?/, parallel io h g ; bisect /?^ in Z, and trom Z, 
draw / Z, parallel to nj; from ?i, draw n Z, at right angles to J n ; 
on the line, b, make t equal to n t. Then, to obtain a mould 
for the twist going up the flight, proceed as at Fig. 293 ; making 
c in that figure equal to e w in Fig, 297, and the other lines of 
a length and position such as is indicated by the letters of reference 
in each figure. To obtain the mould for the level rail, extend h 
o, {Fig. 297,) to i ; make i equal to/ Z, and join i and c ; make 
e i, (Fig. 298,) equal to ciaX Fig. 297 ; through c, draw c 4^ at 



d 

Fig. 298. 



right angles to ci ; make d c equal to dfaX Fig. 297, and com 
plete the parallelogram, odd; then proceed as in the previous 
cases to find the mould. 

464. — All the moulds obtained by the preceding examples have 
been for round rails. For these, the mould may be applied to 
a plank of the same thickness as the rail is intended' to be, and 
the plank sawed square through, the joints being cut square from 
the face of the plank. A twist thus cut and truly rounded will 
hang in a proper position over the plan, and present a perfect and 
graceful wreath. 

465. — To bore for the balusters of a round rail before round- 
i7ig it. Make the angle, c t, [Fig. 299,) equal to the angle, 
c t, at Fig. 292 ; upon c, describe a circle with a radius equal to 
half the thickness of the rail ; draw the tangent, b c?, parallel to 
t c, and complete the rectangle, e b c?/, having sides tangical to 
the circle ; from c, draw c a, at right angles to c; then, b d 
being the bottom of the rai.1, set a gauge from b to a, and run it 
tie whole length of the stuff; in boring, place the centre of th^ 



34*i 



AMERICAN HOUSE-CARPENTER. 




bit in the gauge-mark at a, and bore in the direction, a c. To dc 
this easily, make chucks as represented in the figure, the bottom 
^d^Q, g A, being parallel to o c, and having a place sawed out, as 
e/, to receive the rail. These being nailed to the bench, the rail 
will be held steadily in its proper place for boring vertically. 
The distance apart that the balusters require to be, on the under 
side of the rail, is one-half the length of the rake-side of the 
pitch-board. 




STAIRS. 



343 



4:66. — To obtain^ by the foregoing principles, the face-mould 
for the twists of a moulded rail upon platform stairs In Fig. 
300, ah c is the plan of a line passing through the centre of 
the rail around the cylinder as before, and the lines above 
it are a vertical section of steps, risers and platform, with 
the lines for the rail obtained as in Fig. 292. Set half the width 
of the rail from b to f and from b to r, and from / and r,.draw/ 
e and r d, parallel to c a At Fig. 301, the centre lines of the 

s d n e 



m/ ^ 


^ 


w 


1 W 


o 1 
c 


// 


\ I ^ 


/l 


g 

Fig. 801. 


J 





rai", ]i c and c w, are obtained as in the previous examples. Make 
c i and c y,each equal to c i at Fig. 300, and draw the lines, i m 
andy g^ parallel to c A; ; make n e and n d equal tone and n doX 
Fig. 300, and draw d o and e Z, parallel to n c; also, through k, 
draw s g, parallel to n c ; then, in the parallelograms, m s d o and 
g s el, find the elliptic curves, d m and e g, according to Art. 
118, and they will define the curves. The line, d e, being drawn 
through n parallel to A; c, defines the joint, which is to be cut 
through the plank vertically. If the rail crosses the platform rather 
steep, a butt joint will be preferable, to obtain wl>ch see Art. 498. 




344 



AMERICAN HOUSE-CARPENTEJl.. 



467. — To apply the mould to the plank. The mould obtained 
according to the last article must be applied to both sides of the 
plankj as shown at Fig. 302. Before applying the mould, the 
edge, e/, must be bevilled according to the angle, c ^ ^, at Fig. 
300 ; if the rail is to be canted up, the edge must be bevilled at 
an obtuse angle with the upper face ; but if it is to be canted 
down^ the angle that the edge makes with the upper face mnst be 
acute. From the spring of the curve, a, and the end, c, draw 
vertical lines across the edge of the plank by applying the pitch- 
board, ah c ; then, in applying the mould to the other side, place 
the points, a and c, at h and/; and, after marking around it, saw 
the rail out vertically. After the rail is sawed out, the bottom 
and the top surfaces must be squared from the sides. 

468. — To ascertain the thickness of stuff required for the 
twists. The thickness of stuff required for the twists of a round 
rail, as before observed, is the same as that for the straight ; but 
for a moulded rail, the stuff for the twists must be thicker than 
that for the straight. In Fig. 300, draw a section of the rail be- 
tween the lines, d r and ef and as close to the line, d e, as possi- 
ble ; at the lower corner of the section, draw g A, parallel to d e; 
then the distance that these lines are apart, will be the thickness 
required for the twists of a moulded rail. 

The foregoing method of finding moulds for rails is applicable 
to all stairs which have continued rails around cylinders, and are 
without winders. 

WINDING STAIRS. 

469. — Winding stairs have steps tapering narrower at one end 
than at the other. In some stairs, there are steps of parallel width 
incorporated with tapering steps ; the former are then caMed flyers 
and the latter tiinders. 

470. — To describe a regular geometrical winding stairs. 
In Fig. 303, abed represents the inner surface of the wall en- 
closing the space allotted to the stairs, a e the length of the steps, 
and efgh the cylinder, or face of the front string. The line, 



STAIRS. 



345 




Fig. 303 



U^ 



a e, is given as the face of the first riser, and the point, j^ for the 
limi: of the last. Make e i equal to 18 inches, and upon o, with 
i for radius, describe the arc, ij; obtain the number of risers 
and of treads required to ascend to the floor aXj, according to Art. 
454, and divide the arc, ij, into the same number of equal parts 
as there are to be treads ; through the points of division, 1, 2, 3, 
&c., and from the wall-string to the front-string, draw lines tend- 
ing to the centre, o ; then these lines will represent the face of 
each riser, and determine the form and width of the steps. Allow 
the necessary projection for the nosing beyond a e, which should 
be equal to the thickness of the step, and then ael k will be the 
dimensions for each step. Make a pitch-board for the wall-string 
having a k for the tread, and the rise as previously ascertained ; 
with this, lay out on a thicknessed plank the several risers and 
treads, as at Pig. 287, gauging from the upper edge of the string 
for the line at which to set the pitch-board. 

Upon the back of the string, with a IJ inch dado plane, make 

44 



me 



AMERICAN HOUSE-CARPENTER. 



a succession of grooves IJ inches apart, and parallel with the 
lines for the risers on the face. These grooves must be cut along 
the whole length of the plank, and deep enough to admit of the 
plank's bending around the curve, abed. Then construct a 
drum, or cylinder, of any common kind of stuff, and made to fit 
a curve having a radius the thickness of the string less than o a ; 
upon this the string must be bent, and the grooves filled with strips 
of wood, called keys^ which must be very nicely fitted and glued 
in. After it has dried, a board thin enough to bend around on the 
outside of the string, must be glued on from one end to the other 
and nailed with clout nails. In doing this, be careful not to nail 
into any place where a riser or step is to enter on the face. 

After the string has been on the drum a sufficient time for the 
glue to set, take it off, and cut the mortices for the steps and 
risers on the face at the lines previously made ; which may be 
done by boring with a centre-bit half through the string, and 
nicely chiseling to the line. The drum need not be made so 
large as the whole space occupied by the stairs, but merely large 
enough to receive one piece of the wall-string at once — for it 
is evident that more than one will be required. The front string 
may be constructed in the same manner ; taking e I instead of a 
k for the tread of the pitch-board, dadoing it with a smaller dado 
plane, and bending it on a drum of the proper size. 



a III no y 



Fig. 804. 

471. — To find the shape and position of the timbers 7ieceS' 
sary to support a winding stairs. The dotted lines in Fig. 
303 show the proper position of the timbers as regards the plan : 
the shape of each is obtained as follows. In Fig. 304, the line, 
1 a, is equal to a riser, less the thickness of the floor, and the 
lines, 2 m, 3 n, 4 0, 5 p and 6 ^, are each equal to one riser. The 



STAIRS. 347 

line, a 2, is equal to a m in Pig. 303, the line, m 3 to m n in that 
figure, (fee. In drawing this figure, commence at a, and make 
the lines, a 1 and a 2, of the length above specified, and draw 
them at right angles to each other ; draw 2 m, at right angles to 
a 2, and m 3, at right angles to m 2^ and make 2 m and m 3 of 
the lengths as above specified ; and so proceed to the end. Then, 
through the points, 1, 2, 3, 4, 5 and 6, trace the line, lb; upon 
the points, 1, 2, 3, 4, (fee, with the size of the timber for radius, 
describe arcs as shown in the figure, and by these the lower line 
may be traced parallel to the upper. This will give the proper 
shape for the timber, a b, in Fig. 303 ; and that of the others may 
be found in the same manner. In ordinary cases, the shape of 
one face of the timber will be sufficient, for a good workman 
can easily hew it to its proper level by that ; but where great 
accuracy is desirable, a pattern for the other side may be found 
in the same manner as for the first. 

472. — To find the falling-mould for the rail of a winding 
stairs. In Fig. 305, a cb represents the plan of a rail around 
half the cylinder, A the cap of the newel, and 1, 2, 3, (fee, the 
face of the risers in the order they ascend. Find the stretch-out, 
e /, of a c 6, according to Art. 92 ; from o, through the point of 
the mitre at the newel-cap, draw o s ; obtain on the tangent, e c?, 
the position of the points, s and K\* as at t and/^ ; from e tp and 
/; draw e x,t u^f g^ ^vA f h^ all at right angles to e d ; make e 
g equal to one rise and/^^^ equal to 12, as this line is drawn 
from the 12th riser ; from g^ through g^^ draw^ i; make g x equal 
to about three-fourths of a rise, (the top of the newel, x^ should 
be 3 J feet from the floor ;) draw x u, at right angles to e x, and 
ease off the angle at w ; at a distance equal to the thickness of 

* In the above, the references, a^, b'^, &c., are introduced for the first time. During the 
lime taken to refer to the figure, the memory of the form of these may pass from the mind, 
while that of the sound alone remains ; they may then be mistaken for a 2, 6 2, &c. This 
«m be avoided in reading by giving them a sovnd corresponding to their meaning, which 
ii second a second b, &c. or a second, b second. 



AMERICAN HOUSE-CARPENTER. 




Fig. 305. 



the rail, draw v wy^ parallel to xui; from the centre of the plan, 
0, draw o I, at right angles to e d ; bisect hn in p, and through 
p, at right angles to g i, draw a line for the joint ; in the same 
manner, draw the joint at k ; then a: y will be the falling-mould 
for til at part of the rail which extends from 5 to 6 on the plan. 

473. — To find tl,e face-mould for the rail of a winding-stairs. 
From the extremities of the joints in the falling-mould, as Ar, z 
and y, (Fig. 305,) draw k d., z ¥ and y d, at right angles to e d ; 
make b ^ equal to / d. Then, to obtain the direction of tho 
joint, c^ c', or 6^ c?', proceed as at Fig 306, at which the parts are 



STAIRS. 



349 




Fig. 306. 



shown at half their full size. A is the plan of the rail, and B is 
the falling-mould ; in which k z is the direction of the butt-joint. 
From k, draw k 6, parallel to I o, and k e, at right angles to k b ; 
from 6, draw b /, tending to the centre of the plan, and from/, draw 
/ e, parallel iob k ; from /, through e draw I i, and from ^, draw i 
dj parallel toef; join d and 6, and d b will be the proper direction 



350 



AMERICAN HOUSE-CARPENTER. 



for the joint on the plan. The direction of the joint on the other 
side, a c, can be found by transferring the distances, x b and o d 
to X a and o c. (See Art. 4Y7.) 




Fig. 307. 



Having obtained the direction of the joint, make s r d b, {Fig. 
307,) equal to s r d' 6' in Fig, 305 ; through r and d, draw t a , 
through s and from d, draw t u and d e, at right angles to ^ a / 
make t u and d e equal to t u and 6^ m, respectively, in Fig. 305 ; 
from w, through e, draw u o ; through ft, from r, and from as many 
other points in the line, t a, as is thought necessary, as/, h and J 
draw the ordinates, r c^f g^h i,j k and ao ; from w, c, ^, % k, e 
and 0, draw the ordinates, w 1, c 2, ^ 3, i 4, A' 5, e 6 and o 7, at 
right angles to u o ; make w 1 equal to t s, c 2 equal to r 2, ^ 3 
equal to/ 3, <fec., and trace the curve, 1 7, through the points 
thus found ; find the curve, c e, in the same manner, by transfer- 
ring the distances between the line, t a, and the arc, r d ; join 1 
and c, also e and 7 ; then, 1 c e 7 will be the face-mould required 
for that part of the rail which is denoted by the letters, s r d^ h^, 
on the plan at Fig. 305. 

To ascertain the mould for the next quarter, make acje, (Fig 



BTAIRS. 



351 




Fig. 808. 



308,) equal to a' c^j e^ at Fig. 305 ; at any convenient height on 
the line, d i, in that figure, draw q i\ parallel to e d ; through c 
and J, {Fig. 308,) draw b d ; through a, and from 7, draw b k and 
; 0, at right angles to b d ; make b k and J equal to ^ k and q 
i, respectively, in Fig. 305 ; from k^ through 0, draw kf; and 
proceed as in the last figure to obtain the face-mould, A. 

474. — To ascertain the requisite thickness of stuff. Case 
1. — When the falling-mould is straight. Make h and k m, 
{Fig. 308,) equal to i y at Fig. 305 ; draw h i and m n, parallel 
tob d ; through the corner farthest from kfa,sn or i, draw n i, 
parallel to kf ; then the distance between kf and n i will give 
the thickness required. 

475. — Case 2. — When the falling-mould is curved. In Fig, 
309, sr dbis equal tosr d^ b'^in Fig. 305. Make a c equal to the 
stretch-out of the arc, s b, according to Art. 92, and divide a c and 
s b. each into a like number of equal parts ; from a and c, and from 
each point of division in the line, a c, draw a k, e I, <fcc., at right an- 
gles to a c , make a A: equal to ^ w in Fig. 305, and c /equal toft* wi 



352 



AMERICAN HOUSE-CARPENTER. 




a e f g h i c 



Tig. SCO 



in that figure, and complete the tailing-mould, kj^ every way equal 
to um in Fig. 305 ; from the points of division in the arc, 56, draw 
lines radiating towards the centre of the circle, dividing the arc. 
r d, in the same proportion as 5 6 is divided ; from d and b, draw 
d t and b ii, at right angles to a c?, and from j and v, draw J u and v 
w, at right angles ioj c ; then x t uw will be a vertical projection 
of the joint, d b. Supposing every radiating line across s r d b — 
corresponding to the vertical lines across k j — to represent a joint, 
find their vertical projection, as at 1, 2, 3, 4, 5 and 6 ; through the 
corners of those parallelograms, trace the curve lines shown in the 
figure ; then 6 u will be a helmet, or vertical projection, of s r d b. 
To find the thickness of plank necessary to get out this part of 
the rail, draw the line, z t, touching the upper side of the helinet 
in two places : through the corner farthest projecting from that 
line, as w, draw y w, parallel to z t ; then the distance between 
those lines will be the proper thickness of stuff for this part of the 
rail. The same process is necessary to find the thickness of 
stuff in all cases in which the falling-mould is in any way curved. 
4Y6. — To apply the face-mould to the plank. In Fig. 310, 
A represents the plank with its best side and edge in view, and 
B the same plank turned ap so as to bring in view the other side 



STAIRS. 



353 




Fig. 310. 



and the same edge, this being square from the face. Apply the 
tips of the mould at the edge of the plank, as at a and o, (^5) and 
mark out the shape of the twist ; from a and 0, draw the lines, a 
b and c, across the edge of the plank, the angles, e a b and e 
c, corresponding with kfddX Fig. 308 ; turning the plank up as 
at B, apply the tips of the mould at b and c, and mark it out as 
shown in the figure. In sawing out the twist, the saw must be 
be moved in the direction, ab ; which direction will be perpen- 
dicular when the twist is held up in its proper position. 

In sawmg by the face-mould, the sides of the rail are obtained ; 
the top and bottom, or the upper and the lower surfaces, are ob- 
tained by squaring from the sides, after having bent the falling- 
mould around the outer, or convex side, and marked by its edges. 
Marking across by the ends of the falling-mould will give the 
position of the butt-joint. 

47T. — Elucidation of the process by which the direction of 
the butt-joint is obtained in Art. 473. Mr. Nicholson, in his 
Carpenter'' s Guide., has given the joint a different direction to 
that here shown ; he radiates it towards the centre of the cylin- 
der. This is erroneous — as can be shown by the following 
operation : 

In Fig. 311, a rj i is the plan of a part of the rail about the 
joint, s uis the stretch-out of a i, and gp is the helinet, or ver- 
tical projection of the plan, a r j i, obtained according to Art 

45 



354 



AMERICAN HOUSE-CARPENTER. 




Fig. 811. 



4Y5. Bisect r t^ part of an ordinate from the centre of the plan, 
and through the middle, draw c 6, at right angles to g v ; from 
b and c, draw c d and b e, at right angles to s u ; from d and e, 
draw lines radiating towards the centre of the plan : then d o 
and e m will be the direction of the joint on the plan, according to 
Nicholson, and c b its direction on the falling-mould. It will be 
admitted that all the lines on the upper or the lower side of the rail 
which radiate towards the centre of the cylinder, as d o, e m or 
ij, are level ; for instance, the level line, w v, on the top of the 



STAIRS. 



355 



rail in the helinet, is a true representation of the radiating line, j i^ 
on the plan. The line, b h, therefore, on the top of the rail in 
the helinet, is a true representation of e m on the plan, and A; c on 
the bottom of the rail truly represents d o. From k, draw k l, 
parallel to c 6, and from h, draw hf, parallel to b c ; join I and 
b, also c and/; then c k I b will be a true representation of the 
end of the lower piece, B, and c fh b of the end of the upper 
piece, A ; and/ k or h I will show how much the joint is open on 
the inner, or concave side of the rail. 




Fig. 812. 



356 



AMERICAN HOUSE -CARPENTER. 



To show that the process followed in Art. 473 is correct, let d o 
and e m, [Fig. 312,) be the direction of the butt-joint found as at 
Fig. 306. Now, to project, on the top of the rail in the helinet, a 
line that does not radiate towards the centre of the cylinder, as^' 
A:, draw vertical lines from 7 and k to w and h, and join w and h ; 
then it will be evident that wh is a true representation in the helinet 
of j k on the plan, it being in the same plane as ; A:, and also in the 
same winding surface as w v. The line, I n, also, is a true repre- 
sentation on the bottom of the helinet of the line, J k, in the plan. 
The line of the joint, e m, therefore, is projected in the same way 
and truly by i 6 on the top of the helinet ; and the line, d 0, by 
c a on the bottom. Join a and i, and then it will be seen that 
the lines, c a, a i and i b, exactly coincide with c b, the line of 
he joint on the convex side of the rail ; thus proving the lower 
L-nd of the upper piece, Aj and the upper end of the lower piece, 
B, to be in one and the same plane, and that the direction of the 
joint on the plan is the true one. By reference to Fig. 306 it will 
be seen that the line, I i, corresponds to ;r i in Fig. 312 ; and 
that e k in that figure is a representation of/ b, and i k oi db. 





Fig. 818. 



In getting out the twists, the joints, before the falling-mould is 



STAIRS 357 

applied, are cut perpendicularly, the facfc-mouiu being longenodgh 
to include the overplus necessary for a butt-joint. The face-mould 
for A, therefore, would have to extend to the line, i b ; and that for 
B, to the line, y z. Being sawed vertically at first, a section of the 
joint at the end of the face-mould for A, would be represented in 
the helinet hybifg. To obtain the position of the line, b i, on 
the end of the twist, draw i s, {Fig. 313,) at right angles to ij] 
and make i s equal to m e at Fig. 312 ; through s, draw s g, pa- 
rallel to i/, and make s b equal to 5 6 at Fig. 312 ; join b and i ; 
make i/equal to i /at Fig. 312, and from/, draw fg, parallel to \ 
b ; then i b gf will be a perpendicular section of the rail over tht 
line, e m, on the plan at Fig. 312, corresponding toi b gf in the 
helinet at that figure ; and when the rail is squared, the top, or 
back, must be trimmed off to the line, i b, and the bottom to the 
line,/^. 

478. — To grade the front string of a stairs, having winders 
in a qvarter-circle at the top of the flight connected with flyers 
at the bottom. In Fig. 314, a b represents the line of the facia 
along the floor of the upper story, bee the face of the cylinder, 
and c d the face of the front string. Make^ b equal to ^ of the 
diameter of the baluster, and draw the centre-line of the rail,/ ^, 
g h i and ij, parallel to a b, b e c and c d; make g k and g I 
each equal to half the width of the rail, and through k and Z, 
draw lines for the convex and the concave sides of the rail, parallel 
to the centre-line ; tangical to the convex side of the rail, and parallel 
to k m, draw no; obtain the stretch-out, q r, of the semi-circle, k 
p nij according to Art. 92 ; extend ab to t, and k mto s; make c 5 
equal to the length of the steps, and i u equal to 18 inches, and de- 
scribe the arcs, s t and u 6, parallel to m p ; from t, draw t w^ tend- 
ing to the centre of the cylinder ; from 6, and on the line, ^ux, run 
off the regular tread, as at 5, 4, 3. 2, 1 and v ; make u x equal to 
half the arc, u 6, and make the point of division nearest to x^ as 
27, the limit of the parallel steps, or flyers ; make r o equal Xomz ; 
from 0, draw o a\ at right angles to n o, and equal to one rise ; 



AMERICAN HOUSE-CARPENTEB. 




Fig. 814. 



from c^^ draw a^ s, parallel to 7i o, and equal to one tread ; from s 
through 0, draw s U^. 

Then from w^ draw w c^, at right angles to n o, and set up, on 
the line, w c", the same number of risers that the floor. A, is above 
the first winder, jB, as at 1, 2, 3, 4, 5 and 6; through 5, (on the 
arc, 6 w,) draw d^ e', tending to the centre of the cylinder ; from 
e^, draw e'/^, at right angles to n o, and through 5, (on the line, 



STAIRS. 359 

w i.^,) draw g'^f^, parallel tono ; through 6, (on the line, w cV; 
and/**, draw the line, A^ 6%- make 6 coequal to half a rise, and 
from c^ and 6, draw c^ i^ and 6/, parallel ion o ; make /i^ r equal 
to AV%* ^^^^ A draw f^ ^^, at right angles to i^ A^, and from/'-, 
draw/^A:^, at right angles to/Vi%- upon F, with A:^/^ for radius, 
describe the arc,/^ i%- make 6^ Z^ equal to b'^f% and ease off the 
angle at b"^ by the curve, /^ Z^. In the figure, the curve is de- 
scribed from a centre, but in a full-size plan, this would be imprac- 
ticable ; the best way to ease the angle, therefore, would be with 
a tanged curve, according to Art, 89. Then from 1, 2, 3 and 4, 
(on the line, w c"^,) draw lines parallel to 7i o, meeting the curve in 
7rf, 7i^, 0^ and // ; from these points, draw lines at right angles to 
n 0, and meeting it in ^-, r'^, s"^ and f; from x^ and r\ draw lines 
tending to u^, and meeting the convex side of the rail in y^ and 
z'^ ; make m v^ equal to r 5^, and m i^;^ equal to r f ; from y\ z'\ 
v\ and vj^, through 4, 3, 2 and 1, draw lines meeting the line of 
the wall -string in a^, b^, c^ and d^ ; from e^, where the centre-line of 
the rail crosses the line of the floor, draw e^/^, at right angles to ii 
0, and from/^, through 6, draw/^ g^ ; then the heavy lines,/^^^, 
e' (P, y^ a^, z^ 6^, v^ c^, w^ d^^ and z y, will be the lines for the risers, 
which, being extended to the line of the front string, b e c d, will 
give the dimensions of the winders, and the grading of the front 
string, as was required. 

479. — To obtain the falling-mould for the twists of the last- 
mentioned stairs. Makei^^^ and f h\ (Fig. 314,) each equal 
to half the thickness of the rail ; through h^ and g^, draw h^ r 
a,nd g'^f, parallel to i^ s ; assuming k k^ and m ttv' on the plan as 
the amount of straight to be got out with the twists, make n q 
equal to k k^, and r P equal to m m%- from n and l^, draw lines at 
right angles to n o, meeting the top of the falling-mould in n^ and 
0^ ; from o^, draw a line crossing the falling-mould at right angles 
to a chord of the curve, /^ P ; through the centre of the cylinder, 
draw u^ 8, at right angles to no ; through 8, draw 7 9, tending tc 
k"^ ; then w' 7 will be the falling-mould for the upper twist, and 7 
0^ the falling-mould for the lower twist. 



360 



AP.IERICAN HOUSE-CARPENTER. 



480. — To obtain the face-moulds. The moulds for the twists 
of this stairs may be obtained as at Art. 473 ; but, as the falhng- 
mould in its course departs considerably from a straight line, it 
would, according to that method, require a very thick plank for 
the rail, and consequently cause a great waste of stuff. In order, 
therefore, to economize the material, the following method is to 
be preferred — in which it will be seen that the heights are taken 
in three places instead of two only, as is done in the previous 
method. 




Fig. 315. 



Case 1. — When the middle height is above a line joining 
the other two. Having found at Fig. 314 the direction of the 
joint, %o s^ and p e, according to Art. 4Y3, make k p e a, {Fig. 
315,) equal to k^ jy^ e p in Fig. 314 ; join b and c, and from o, 
draw ^, at right angles to 6 c ; obtain the stretch-out of d g, as 
df, and at Fig. 314, place it from the axis of the cylinder, jj, to 
(f ; from q^ in that figure, draw cf r^, at right angles ton o ; also, 
at a convenient height on the line, 7i 7i^, in that figure, and at 
right angles to that line, draw u^ v^ ; from b and c, in Fig. 315, 



STAIRS. 361 

draw 6 J and c Z, at right angles to b c ; make bj equal to w' n^ in 
Fig. 314, i h equal to i^?^ r^ in thn.t figure, and c I equal to v^ 9 ; 
from I, through J, draw Im ; from A, draw h n, parallel to c b ; 
from n, draw tz r, at right angles to b c, and join r and s ; through 
the lowest corner of the plan, as jo, draw v e, parallel to b c ; from 
a, e, i«, p, k, t, and from as many other points as is thought ne- 
cessary, draw ordinates to the base-line, v e, parallel to r s ; 
through A, draw w x^ at right angles to m I ; upon n, with r s for 
radius, describe an intersecting arc at x, and join n and x; from 
the points at which the ordinates from the plan meet the base- 
line, V e, draw ordinates to meet the line, ?n I, at right angles to v 
6 ; and from the points of intersection on m I, draw correspond- 
ing ordinates, parallel to n x ; make the ordinates which are pa- 
rallel to /i ^ of a length corresponding to those which are parallel 
to r 5, and through the points thus found, trace the face-mould 
as required. 

Case 2. — When the middle height is below a line joining 
the other tivo. The lower twist in Fig. 314 is of this nature. 
The face-mould for this is found at Fig. 316 in a manner similar 
to that at Fig. 315. The heights are all taken from the top of 
the falling-mould at Fig. 314 ; b j being equal to i^? 6 in Fig. 314, 
i h equal to x^ y^ in that figure, and c Z to f &. Draw a Ime 
through^' and Z, and from A, draw h n^ parallel \o b c ; from n^ 
draw n r, at right angles to b c, and join r and s ; then r s will be 
the bevil for the lower ordinates. From A, draw A x, at right an- 
gles toj I ; upon n, with r s for radius, describe an intersecting 
arc at x, and join n and x ; then n x will be the bevil for the upper 
ordinates, upon which the face-mould is found as in Case 1. 

481. — Elucidation of the foregoing method. — This method 
of finding the face-moulds for the handrailing of winding stairs, 
being founded on principles which govern cylindric sections, may 
be illustrated by the following figures. Fig. 317 and 318 repre- 
sent solid blocks, or prisms, standing upright on a level base, b d ; 
the upper surface,^* a forming oblique angles with the face, b I — 

46 



362 



AMERICAN HOUSE-CARPENTEa. 




Fig. 316. 



ill Fig. 317 obtuse, and in Fig. 318 acute. Upon the base, de- 
scribe the semi-circle, h s c ; from the centre, i, draw i 5, at right 
angles to 6 c ; from 5, draw 5 x^ at right angles to e c?, and from i 
draw i A, at right angles ioh c ; make i h equal to s x, and join 
h and a:; then, h and x being of the same height, the line, h x, 
joining them, is a level line. From A, draw h n, parallel to b c, 
and from 7^, draw n r, at right angles to 6 c ; join r and s, also w 



ATAIBS. 



36: 





Fig. 317. 



Fig-. 81S. 



and ^; then, ii and x being of the same height, nx\s?i level Hne ; 
and this Une lying perpendicularly over r s, n x and r s must be 
of the same length. So, all lines on the top, drawn parallel to 7i 
Xj and perpendicularly over corresponding lines drawn parallel to 
r s on the base, must be equal to those lines on the base ; and by 
drawing a number of these on the semi-circle at the base and 
others of the same length at the top. it is evident that a curve, j 
X Z, may be traced through the ends of those on the top, which 
shall lie perpendicularly over the semi-circle at the base. 

It is upon this principle that the process at Fig. 315 and 316 
is founded. The plan of the rail at the bottom of those figures 
is supposed to lie perpendicularly under the face-mould at the top ; 
and each ordinate at the top over a corresponding one at the base. 
The ordinates, n x and r s, in those figures, correspond to n x 
and r s in Fig. 317 and 318. 

In Fig. 319, the top, e «, forms a right angle with the face, d 
c ; all that is necessary, therefore, in this figure, is to find a line 
corresponding to A a; in the last two figures, and that will lie level 
and in the upper surface ; so that all ordinates at right angles to 
d r on the base, will correspond to those that are at right angles 



364 



AMERICAN HOUSE-CARPENTER. 




Fig 319 r 



10 e c on the top. This elucidates Fig. 307 ; at which the lines, 
h 9 and i 8. correspond to li 9 and i 8 in this figure. 




Fig. 320. 



482. — To find the hevil for the edge of the 'plank. The 
plank, before the face-mould is applied, must be bevilled accord- 
ing to the angle which the top of the imaginary block, or prism, 
in the previous figures, makes with the face. This angle is de- 
termined in the following manner : draw w i, {Fig' 320,) at right 
angles to i s, and equal to wh a.t Fig. 315 ; make i s equal to t 5 in 
that figure, and join w and s ; then sw p w^ill be the bevil required 
m order to apply the face-mould at Fig. 315. In Fig. 316, the 
middle height being below the line joining the other two, the bevil 
is therefore acute. To determine this, draw i s, {Fig. 321,) al 



STAIRS. 



365 




Fig. 821. 

right angles to ip^ ai d equal to ^ 5 in Fig, 316 ; make 5 u equal 
to h w in Fig. 316, and join w and i ; then w i p will be the 
bevil required in ordsr to apply the face-mould at Fig. 316. Al 
though the falling-mould in these cases is curved, yet, as the 
plank is sprung, or bevilled on its edge, the thickness necessary 
to get out the twist may be ascertained according to Art. 474 — 
taking the vertical distance across the falling-mould at the joints, 
and placing it down from the two outside heights in Fig. 315 or 
316. After bevilling the plank, the moulds are applied as at Art. 
476 — applying the pitch-board on the bevilled instead of a square 
edge, and placing the tips of the mould so that they will bear the 
same relation to the edge of the plank, as they do to the line, j I, 
in Fig. 315 or 316. 




Fig. 822. 



483.— To apply the moulds without bevilling the plank. 
Make w p, {Fig. 322,) equal to w p aX Fig. 320, and the angle, 
bed, equal to 6 j I in Fig. 315 ; make p a equal to the thick- 
ness of the plank, as w a ir. Fig. 320, and from a draw a 0. pa- 
rallel low d; from c, draw c e, at right angles to w d, and join c 



AMERICAN HOUSE-CARPENTER. 

and b ; then the angle, 6 e o, on a square edge of the plank, hav 
ing a line on the upper face at the distance, p a, in Fig. 320, at 
which to apply the tips of the mould — will answer the same pur- 
pose as bevilling the edge. 

If the bevilled edge of the plank, which reaches from p to Wy 
is supposed to be in the plane of the paper, and the point, a, to 
be above the plane of the paper as much as a, in Fig, 320, is dis- 
tant from the line, wp ; and the plank to be revolved on p 6 as 
an axis until the line, p w, falls below the plane of the paper, and 
the line, p a, arrives in it ; then, it is evident that the point, c, 
will fall, in the line, c e, until it lies directly behind the point, e, 
and the line, b c, will lie directly behind b e. 

k 




Fig. 823. 



484. — To find the b evils for splayed work. The principle 
employed in the last figure is one that will serve to find the bevils 
for splayed work — such as hoppers, bread-trays, (fee. — and a way 
of applying it to that purpose had better, perhaps, be introduced 
in this connection. In Fig. 323, a 6 c is the angle at which the 
work is splayed, and b c?, on the upper edge of the board, is at 
right angles to a b ; make the angle, /^ J, equal to ab c, and 
from/, draw/^, parallel to e a; from 6, draw b o, at right an. 
gks toa b ; through o, draw i e, parallel to c b, and join e and 
d ; then the angle, a e d^ will be the proper bevil for the ends from 
the inside, or k d e from the outside. If a mitre-joint is re- 



STAIRS. 367 

quired, seifg^ the thickness of the stuff on the level, from e to 
m, and join m and d ; then k d m will be the proper bevil for a 
mitre-joint. 

If the upper edges of the splayed work is to he bevilled, so as 
to be horizontal when the work is placed in its proper position, 
fgj, being the same as a 6 c, will be the proper bevil for that 
purpose. Suppose, therefore, that a piece indicated by the lines, 
k g, gf and /A, were taken off; then a line drawn upon the 
bevilled surface from d, at right angles to k d, would show the 
true position of the joint, because it would be in the direction of 
the board for the other side ; but a line so drawn would pass 
through the point, o, — thus proving the principle correct. So, if 
a line were drawn upon the bevilled surface from c?, at an angle 
of 45 degrees to k d, it would pass through the point, n. 

485. — Another method for face-moulds. It will be seen by 
reference to Art. 481, that the principal object had in view in the 
preparatory process of finding a face-mould, is to ascertain upon it 
the direction of a horizontal line. This can be found by a method 
different from any previously proposed ; and as it requires fewer 
lines, and admits of less complication, it is probably to be preferred. 
It can be best introduced, perhaps, by the following explanation . 

In Fig. 324, J d represents a prism standing upon a level base, 
b d, its upper surface forming an acute angle with the face, 
b /, as at Fig. 318. Extend the base line, b c, and the raking 
line, J I, to meet at/; also, extend e d and ^ a, to meet at k; 
from /, through A*, draw / m. If we suppose the prism to stand 
upon a level floor, ofm, and the plane, ^* g a I, to be extended 
to meet that floor, then it will be obvious that the intersection 
Detween that plane and the plane of the floor would be in the line, 
f k; and the line,/ A:, being in the plane of the floor, and also in 
the inclined -plane, jgkf any line made in the plane, J ^ A*/ 
parallel to/ A:, must be a level line. By finding the position of a 
perpendicular plane, at right angles to the raking -plane, jf k g, 
we shall greatly shorten the process for obtaining ordinates. 



363 



AMERICAN HOUSE-CARPENTER. 




Fig. 824. 



This may be done thus : fromjT, draw/ o. at right angles io fm ; 
extend e 6 to o, and^ J, to t ; from o, draw o t, at right angles to 
of, and join t and/; then t of will be a perpendicular plane, at 
right angles to the inclined plane, t g kf; because the base of 
the former, of is at right angles to the base of the latter,/ k, both 
these lines being in the same plane. From 6, draw h p, at right 
angles to of or parallel to fm ; fromp, draw p q, at right angles 
to of and from q, draw a line on the upper plane, parallel to fm, 
or at right angles to tf; then this line will obviously be drawn 
to the point, ^', and the line, qj, be equal top h. Proceed, in the 
same way, from the points, 5 and c, to find a: and I. 

Now, to apply the principle here explained, let the curve, h s c, 
{Fig. 325,) be the base of a cylindric segment, and let it be re- 
quired to find the shape of a section of this segment, cut by a 
plane passing through three given points in its curved surface : 
one perpendicularly over 6, at the h-sight, hj: one perpendicu- 
larly over 5, at the height, s x ; and the other o^er c, at the height, 
c I — these lines being drawn at right angles to the chord of the 
base, h c. From^*, through /, draw a line to meet the chord line 
extended to/; from 5, draw 5 k, parallel to h f and from x^ 
draw X k, parallel to jf ; from/, through k, dTa,wfm; then fm 
will be the intersecting line of *he plane of the section with the 



STAIRS. 



369 




Fig. 825. 



plane of the base. This Une can be proved to be the intersection 
of these planes in another way ; from 6, through s, and from j, 
through .r, draw lines meeting at m ; then the point, m, will be 
in the intersecting line, as is shown in the figure, and also at 
Fig. 324. 

From/, draw/p, at right angles to/ w; from b and c, and 
from as many other points as is thought necessary, draw ordinates, 
parallel to fm; make p q equal to b j, and join q and/; from 
the points at which the ordinates meet the line, qf, draw others 
at right angles to q f; make each ordinate at A equal to its cor- 
responding ordinate at C, and trace the curve, gn i, through the 
points thus found. 

Now it may be observed that A is the plane of the section, B 
the plane of the segment, corresponding to the plane, q pf, of 
Fig. 324, and C is the plane of the base. To give these planes 
their projer position, let A be turned on q f as an axis until it 



370 AMERICAN HOUSE-CARPENTER. 

Stands perpendicularly over the line, qf^ and at right angles to 
the plane, B ; then, while A and B are fixed at right angles, let 
B be turned on the line, p /, as an axis until it stands perpendicu- 
larly over p/, and at right angles to the plane, C ; then the plane, 
A , will lie over the plane, C, with the several lines on one corres- 
ponding to those on the other ; the point, i, resting at Z, the point, 
n, at a;, and g aij ; and the curve, g n i, lying perpendicularly 
over b s c — as was required. If we suppose the cylinder to be 
cut by a level plane passing through the point, I, (as is done in 
finding a face-mould,) it will be obvious that lines corresponding 
to ^ / and ^/ would meet in I ; and the plane of the section, A, 
the plane of the segment, B, and the plane of the base, C, would 
all meet in that point. 

486. — To find the face-mould for a hand-rail according to 
the principles explained in the previous article. In Fig. 326, 
a e cf is the plan of a hand-rail over a quarter of a cylinder ; and 
in Fig. 327, a b c d is the falling-mould ; / e being equal to the 
stretch-out of a df in Fig. 326. From c, draw c h, parallel to 
ef; bisect h c in i, and find a point, as 6, in the arc, df (Fig. 
326.) corresponding to i in the line, he; from i, {Fig. 327,) to 
the top of the falling-mould, draw ij,^t right angles to Ac; at Fig. 
326, from c, through b, draw c g. and from b and c, draw bj and 
c k, at right angles to ^ c ; make c k equal to h g at Fig. 327, 
and bj equal to ij at that figure ; from k, through J, draw k g, 
and from^, through a, draw g p ; then ,£^/) will be the intersecting 
line, corresponding to fm in Fig. 324 and 325 ; through e, draw 
p 6, at right angles to gp, and from c, draw c q. parallel to gp ; 
make r q equal to A ^ at Fig. 327 ; joinp and q, and proceed as 
in the previous examples to find the face-mould, A. The joint 
of the face-mould, u v, will be more accurately determined by 
finding the projection of the 'centre of the plan, o, as at to ; 
joining s and w, and drawing u v, parallel to s w. 

It may be noticed that c k and b j are not of a length corres- 
ponding to the above directions : they are but ^ the length given. 



STAIRS. 



371 




Tig. 326. 



3T2 



AMERICAN HOUSE ^ARPENTER, 




Fig.82T. 



The object of drawing these lines is to find the point, g^ and that 
can be done by taking any proportional parts of the lines given, 
as well as by taking the whole lines. For instance, supposing c 
k and h j to be the full length of the given lines, bisect one in i 
and the other in m; then a line drawn from w, through i, will 
give the point, g^ as was required. The point, g^ may also be 



STAIRS. 



373 



obtained thus : at Fig, 32 Y, make h I equal to c 6 in Fig, 326 
rrom /, draw I k^ at right angles to he; from J, draw^* A:, parallel 
to h c ; from g, tlirough Ar, draw g ii ; at Fig. 326, make b g 
equal to Z w in Fig. 327 ; then ^ will be the point required. 

The reason why the points, a, b and c, in the plan of the rail ai 
Fig. 326, are taken for resting points instead of e, i and/, is this : 
the top of the rail being level, it is evident that the points, a and e, 
in the section a e, are of the same height ; also that the point, i, is ot 
the same height as 6, and c as/. Now, if a is taken for a point 
in the inclined plane rising from the line g pj e must be below 
that plane ; if 6 is taken for a point in that plane, i must be below 
it ; and if c is in the plane,/ must be below it. The rule, then, 
for taking these points, is to take in each section the one that is 
nearest to the line, g p. Sometimes the line of intersection, g p, 
happens to come almost in the direction of the line, e r : in such 
case, after finding the line, see if the points from which the 
heights were taken agree with the above rule ; if the heights 
were taken at the wrong points, take them according to the rule 
above, and then find the true line of intersection, which will not 
vary much from the one already found. 




Fig. 828. 



487.— To appl]/ the face-mould thus found to the plank. 
The face-mould, when obtained by this method, is to be applied 
to a square-edged plank, as directed at Art. 476, with this differ- 
ence : instead of applying both tips of the mould to the edge of 



374 AMERICAN HOUSE-CARPENTER. 

the plank, one of them is to be set as far from the edge of the 
plank, as ^, in Fig. 326, is from the chord of the section p q — eus 
is shown at Fig. 328. Jl, in this figure, is the mould applied on 
the upper side of the plank, B^ the edge of the plank, and C, the 
mould applied on the under side ; a b and c d being made equal 
to ^ :r in Fig. 326, and the angle, e a c, on the edge, equal to the 
angle, p q r, at Fig. 326. In order to avoid a waste of stuff, it 
would be advisable to apply the tips of the mould, e and b, im- 
mediately at the edge of the plank. To do this, suppose the 
moulds to be applied as shown in the figure ; then let A be re- 
volved upon e until the point, b, arrives at g, causing the line, e b, 
to coincide with e g : the mould upon the under side of the 
plank must now be revolved upon a point that is perpendicularly 
beneath e, as /; from/, draw / h, parallel to i d, and from d, 
draw d h, at right angles to i d ; then revolve the mould, C, upon 
/, until the point, /i, arrives at j, causing the line,/ A, to coincide 
with fj, and the line, i d, to coincide with k I ; then the tips of 
the mould will be at k and I. 

The rule for doing this, then, will be as follows : make the an- 
gle, ifk, equal to the angle q v x, at Fig. 326 ; make /A; equal 
to /i, and through A:, draw ^ Z, parallel to ij; then apply the 
corner of the mould, i, at ^, and the other corner rf, at the line, k I. 

The thickness of stuff is found as at Art. 474. 

4S8. — To regulate the application of the falling-mould . 
Obtain, on the line, k c, {Fig. 327,) the several points, r, q^p^ I 
and m, corresponding to the points, 6^, a^^ z^ y, &c., at Fig. 326 ; 
from r qp, &c., draw the lines, r t, q u^p v^ &:.c., at right angles 
to he; make h s, r t, q u, <fec., respectively equal to 6 c\ r q^ 5 
d\ (fee, at Fig. 326 ; through the points thus found, trace the 
curve, s w c. Then get out the piece, g s c, attached to the fall- 
ing-mould at several places along its length, as at z, z, z, (fee. 
In applying the falling-mould with this strip thus attached, the 
edge, sw c, will coincide with the upper surface of the rail piece 



STAIRS. 



;75 



before it is squared ; and thus show the proper posiiioii of the fall- 
ing-mould along its whole length. (See Art. 496.) 

SCROLLS FOR HAND-RAILS. 

4:»9. — General rule for finding the size and position of the 
regulating- square. The breadth which tlie scroll is to occupy, 
the number of its revolutions, and the relative size of the regula 
ting square to the eye of the scroll, being given, multiply the 
number of revolutions by 4, and to the product add the number 
of times a side of the square is contained in the diameter of the 
eye, and the sum will be the number of equal parts into which 
the breadth is to be divided. Make a side of the regulating 
square equal to one of these parts. To the breadth of the scroll 
add one of the parts thus found, and half the sum will be the 
length of the longest ordinate. 



6 \ b 

8 
7 

4 



a h 

Fig. 329 



490. — To find the proper centres in the regulating square. 
Let a 2 1 6, {Fig. 329,) be the size of a regulating square, found 
according to the previous rule, the required number of revolu- 
tions being If. Divide two adjacent sides, as « 2 and 2 1, into 
as many equal parts as there are quarters in the number of revo- 
lutions, as seven ; from those points of division, draw lines across 
the square, at right angles to the lines divided ; then, 1 being the 
first centre, 2, 3, 4, 5, 6 and 7, are the centres for the other quar- 
ters, and 8 is the centre for the eye ; the heavy lines that deter- 



376 



AMERICAN HOUSE-CARPENTER 



mine these centres being each one part less in length than its pre 
ceding line. 




Fig. 830. 



491. — To describe the scroll for a hand-rail over a curtail 
step. Let a b, (Fig. 330,) be the given breadth, If the given 
number of revolutions, and let the relative size of the regulating 
square to the eye be l- of the diameter of the eye. Then, by the 
rule, If multiplied by 4 gives 7, and 3, the number of times a 
side of the square is contained in the eye, being added, the sum 
is 10. Divide a 6, therefore, into 10 equal parts, and set one from 
b to c ; bisect a c in e ; then a e will be the length of the longest 
ordinate, (1 d orl e.) From a, draw a d, from e, draw e 1, and 
from b, draw 6/, all at right angles to a b ; make e 1 equal to e 
a, and through 1, draw 1 d, parallel to a b ; set b c from 1 to 2, 
and upon 1 2, complete the regulating square ; divide this square 
as at Fig. 329 ; then describe the arcs that compose the scroll, as 
follows: upon 1, describe de; upon 2, describe e/; upon 3, 
describe/^ ; upon 4 describe g h, <fcc. ; make d I equal to the 



STAIRS. 



377 



width of the rail, and upoa 1, describe Im ; upon 2, aescribe m 
w, (fee. ; describe the eye upon 8, and the scroll is completed, 

492. — To describe the scroll for a curtail step. Bisect d Z, 
[Fig, 330,) in o, and make o v equal to i of the diameter of a 
baluster ; make v w equal to the projection of the nosing, and e 
X equal to id I; upon 1, describe w y, and upon 2, describe y z ; 
also upon 2, describe x i ; upon 3, describe t j, and so around to 
z ; and the scroll for the step will be completed. 

493. — To determine the position of the balusters under the 
scroll. Bisect d Z, (Fig. 330,) in o, and upon 1, with 1 o for ra- 
dius, describe the circle, oru; set the baluster at p fair with the 
face of the second riser, c^, and from p, with half the tread in the 
dividers, space off as at o, q, r, s, t, w, <fcc., as far as q^ ; upon 2, 
3, 4 and 5, describe the centre-line of the rail around to the eye 
of the scroll ; from the points of division in the circle, oru, draw 
lines to the centre-line of the rail, tending to the centre of the 
eye, 8 ; then, the intersection of these radiating lines with the 
centre-line of the rail, will determine the position of the balusters, 
as shown in the figure. 




Fig. 881. 



^94t.— To obtain the falling-mould for the raking j^art of the 
scroll. Tangical to the rail at A, {Fig. 330,) draw h k, parallel to d 
a; then k a' will be the joint between the twist and the other part 
of the scroll. Make d e' equal to the stretch-out of d e, and upon d 

48 



378 



AMERICAN HOUSE-CARPENTER. 



e^^ find the position of the point, A:, as at F ; at Fig. 331, make e d 
equal to e^ d in Fig. 330, and d c equal to d <? in that figure ; 
from c, draw c a, at right angles to e c, and equal to one rise ; 
make c 6 equal to one tread, and from 6, through a, draw 6 j , 
bisect a c in Z, and through Z, draw m g^, parallel to e h ; 7n q \^ 
the height of the level part of a scroll, which should always be 
about 3 J feet from the floor ; ease off the angle, r/ifj^ according 
to Art. 89, and draw g w ii, parallel to m a; j, and at a distance 
equal to the thickness of the rail ; at a convenient place for the 
joint, as i, draw i ?i, at right angles to b j ; through n, draw ; A, 
at right angles to eh ; make d k equal to d k^ in Fig. 330, and 
from A-, draw k o, at right angles to eh; at Fig. 330, make d 
W" equal to d h in Fig. 331, and draw W 6^, at right angles to d 
h^ ; then k a^ and h'^ i^ will be the position of the joints on the 
plan, and at Fig. 331, o p and i 7i, their position on the falling- 
mould ; and p o i n, {Fig. 331,) will be the falling-mould re- 
quired. 




Fig. 332. 



495. — To describe the face-mould. At Fig. 330, fromAr, draw 
k r^, at right angles to r^ d : at Fig. 331, make h r equal to A^ r* 
in Fig. 330, and from r, draw r s, at right angles to r h ; from 
the intersection of r 5 with the level line, m q, through i, draw 5 
t ; at Fig. 330, make h^ 6^ equal to ^ ^ in i^i^. 331, and join 6' 
and r^ ; from a'^, and from as many other points in the arcs, a^ I 
and k d, as is thought necessary, draw ordinates to r^ d, at right 
angles to the latter ; make r 5, (Fig. 332,) equal in its length and 
in its divisions to the line, r"^ b'\ in Fig. 330 ; from r, n, o, p, q 



STAIRS. 



379 



and Z, draw the lines, r k^ n d, o a, p e, qf and I c, at right an- 
gles to r b, and eqmal to r^ k, cP s\ p a^^ &c., in Fig. 330 ; 
through the points thus found, trace the curves, k I and a c, and 
complete the face-mould, as shown in the figure. This mould is 
to be applied to a square-edged plank, with the edge, I b, parallel 
to the edge of the plank. The rake lines upon the edge of the 
plank are to be made to correspond to the angle, s t h, in Fig. 
331. The thickness of stuff required for this mould is shown at 
Fig. 331, between the lines s t and u v — u v being drawn pa- 
rallel to 5 ^. 

496. — All the previous examples given for finding face-moulds 
over winders, are intended for moulded rails. For round rails, 
the same process is to be followed with this difference : instead 
of working from the sides of the rail, work from a centre-line. 
After finding the projection of that line upon the upper plane, 
describe circles upon it, as at Fig. 293, and trace the sides of the 
moulds by the points so found. The thickness of stuff for the 
twists of a round rail, is the same as for the straight ; and the 
twists are to be sawed square through. 

Ti f V k 




380 AMERICAN HOUSE-CARPENTER 

497. — To ascertain the form of the newel-cap from a section 
of the rail. Draw a 6, {Fig. 333,) through the widest pait of 
the given section, and parallel to c d ; bisect ab in e, and through 
a, e and 6, draw h iyfg, and kj, at right angles to a b ; at a con- 
venient place on the Ime, fg, as o, with a radius equal to half 
the width of the cap, describe the circle, i j g ; make r I equal 
to e b 01 e a ; join I and j, also / and i ; from the curve, f b, to 
the line, I j, draw as many ordinates as is thought necessan,^ 
parallel to f g ; from the points at which these ordinates meet 
the line, Ij, and upon the centre, o, describe arcs in continuation to 
meet op; from n, t, x, &c., draw n s, t u, &c,, parallel to f g ; 
make n s, t u, &cc., equal to e f iv v, &;c. ; make x y, &c., equal 
to z dj &c. ; make o 2, o 3, &c., equal to on, o t, &c. ; make 2 4 
equal to n s, and in this way find the length of the lines crossing 
m ; through the points thus found, describe the section of the 
newel-cap, as shown in the figure. 




498. — To find the true position of a butt joint for the twists of 
a moulded rail over platform stairs. Obtain the shape of the 
mould according to Art. 466, and make the line a b, Fig. 334, 
equal to a c. Fig. 300 ; from b, draw b c, at right angles to a 6, 
and equal in length to n m, Fig. 300 ; join a and c, and bisect a c 
in ; through o draw e f at right angles to a c, and d k, parallel 
to c b ; make o d and o k each equal to half e h at Fig. 300 ; 
through e and /, draw h i and ^ j, parallel to a c. At Fig. 301, 
make n a equal to e d, Fig. 334, and through g, draw r p, at right 
angles to n c ; then r p will be the true position on the face-mould 
for a butt joint, as was required. The sides must be sawn verti 



STAIRS. 



381 



cally as described at Art. 467, but the joint is to be sawn square 
through the plank. The moulds obtained for round rails, (Art. 
464,) give the line for the joint, when applied to either side of the 
plank ; but here, for moulded rails, th^; line for the joint can be 
obtained from only one side. Whe i the rail is canted up, the 
joint is taken from the mould laid on the upper side of the lowei 
twist, and on the under side of the upper twist ; but when it is 
canted down, a course just the reverse of this is to be pursued. 
When the rail is not canted, either up or down, the vertical joint, 
obtained as at Art. 466, will be a butt joint, and therefore, in such 
a case, the process described in this article will be unnecessary 



NOTE TO ARTICLE 462. 



Platform stairs with a large cylinder. Instead of 
placing the platform-risers at the spring of the cyl- 
inder, a more easy and graceful appearance may be 
given to the rail, and the necessity of canting either 
of the twists entirely obviated, by fixing the place of 
the above risers at a certain distance within the cyl- 
inder, as shown in the annexed cut — the lines indi- 
cating the face of the risers cutting the cylinder at k 
and I, instead of at p and q, the spring of the cylin- 
der. To ascertain the position of the risers, let a & c 
be the pitch-board of the lower flight, and c d e that 
of the upper flight, these being placed so that h c 
and c d shall form a right line. Extend c c to cut 
dein f; draw / g parallel to d h, and of indefinite 
length : draw g o ^i right angles to / g, and equal 
in length to the radius of the circle formed by the 
j^entre of the rail in passing around the cylinder ; 
on as centre describe the semicircle j g i ; make 
h equal to the radius of the cylinder, and describe 
on o the face of the cylinder phq; then extend d h 
across the cylinder, cutting it in I and k — giving the 
position of the face of the risers, as required. To 
find the face-mould for the twists is simple and ob- 
vious : it being merely a quarter of an ellipse, hav- 
ing j for serai-minor axis, and the distance on the 

rake corresponding to o g, on the plan, for the semi-major axis, found thus,— extend « j tc 
meet a f, then from this point of meeting to / is the semi-major axis. 





\ 


d 










f 




c 


e 


A 


IX 


I 




y 








3 


\ 








^ 






T 


i 






n 




\] 


£* 




9 


Ic 


J 






1 







SECTION VIL— SHADOWS. 



499. — The art of drawing consists in representing solids upon 
a plane surface : so that a curious and nice adjustment of lines is 
n ade to present the same appearance to the eye, as does the 
human figure, a tree, or a house. It is by the effects of light, in 
its reflection, shade, and shadow, that the presence of an object is 
made known to us ; so, upon paper, it is necessary, in order that 
the delineation may appear real, to represent fully all the shades 
and shadows that would be seen upon the object itself. In this 
section I propose to illustrate, by a few plain examples, the simple 
elementary principles upon which shading, in architectural sub- 
jects, is based. The necessary knowledge of drawing, prelim- 
inary to this subject, is treated of in the Introduction, from Art. 
1 to 14. 

500. — TTie inclination of the line of shadow. This is always, 
in architectural drawing, 45 degrees, both on the elevation and the 
plan ; and the sun is supposed to be behind the spectator, and 
over his left • shoulder. This can be illustrated by reference to 
Fig. 335, in which A represents a horizontal plane, and B and C 
two vertical planes placed at right angles to each other. A rep- 
resents the plan, C the elevation, and B a vertical projection 
from the elevation. In finding the shadow of th 3 plane, jB, the 



SHADOWS. 



38S 




Fig. 335. 



line, a b, is drawn at an angle of 45 degrees with the horizon, and 
the line, c b, at the same angle with the vertical plane, B. The 
plane, B, being a rectangle, this makes the true direction of the 
sun's rays to be in a course parallel to d b ; which direction has 
been proved to be at an angle of 35 degrees and 16 minutes with 
the horizon. It is convenient, in shading, to have a set-square 
with the two sides that contain the right angle of equal length ; 
this will make the two acute angles each 45 degrees ; and will 
give the requisite bevil when worked upon the edge of the T- 
square. One reason why this angle is chosen in preference to 
another, is, that when shadows are properly made upon the draw- 
ing by it, the depth of every recess is more readily known, since 
the breadth of shadow and the depth of the recess will be equal. 

To distinguish between the terms shade and shadow, it will be 
understood that all such parts of a body as arc not exposed to the 
direct action of the sun's rays, are in shade ; while those parts 
which are deprived of hght by the interpositir/n of other bodies, 
are in shadow. 



38i 



AMERICAN HOUSE-CARPENTER. 



B iii i i iii i i ii iiii iii i i ii wi niii i i mii ii Himii i ii i iiHpiiiH i iiii i iiii iii i ii iliH iii 




' ^pBMiyjpii ff PPi i llllililp 



Fig. 836. 



Fig. 337. 





Fig. 



Fig. 339. 



601. — To find the line of shadow on mouldings and other ho- 
rizontally straight projections. Fig. 336, 337, 338, and 339, 
lepresent various mouldings in elevation, returned at the left, in 
the usual manner of mitreing around a projection. A mere in- 
spection of the figures is sufficient to see how the line of shadow 
is obtained ; bearing in mind that the ray, a b, is drawn from the 
projections at an angle of 45 degrees. Where there is no return 
at the end, it is necessary to draw a section, at any place in the 
length of the mouldings, and find the line of shadow from that. 

502. — To find the line of shadow cast by a shelf. In Fig. 340, 
A is the plan, and B is the elevation of a shelf attached to a wall. 
From a and c, draw a b and c d, according to the angle previously 
directed ; from 6, erect a perpendicular intersecting c dat d ; from 
d, draw d e, parallel to the shelf ; then the lines, c d and d e, will 
define the shadow cast by the shelf. There is another method of 
finding the shadow, without the plan, A. Extend the lower line 
o* tne shelf to/, and make of equal to the projection of the shelf 



SHADOWS. 



385 




Fig. S40. 

from the wall ; from/, draw/^, at the customary angle, and from 
c, drop the vertical line, c g, intersecting f g dX g ; from g, draw 
g e, parallel to the shelf, and from c, draw c d, at the usual angle ; 
then the lines, c d and d e, will determine the extent of the shadow 
as before. 



'"''lIlIilH 



B 




503. — To find the shadow cast hy a shelf , which is wider at 
one end than at the other. In Fig. 341, A is the plan, and B 
the elevation. Find the point, d, as in the previous example, and 
from any other point in the front of the shelf, as a, erect the perpen- 
dicular, a e ; from a and e, draw a h and e c, at the proper angle, 
and from 6, erect the perpendicular, h c, intersecting e c in c ; 

49 



386 



AMERICAN HOrSE-CARPENTER. 



from d, through c, draw d o ; then the Hnes, i d and d o, will give 
the limit of the shadow cast by the shelf. 




Fig. S42. 

504. — To find the shadow of a shelf having one end acute or 
obtuse angled. Fig. 342 shows the plan and elevation of an 
acute-angled shelf. Find the line, e g, as before ; from a, erect 
the perpendicular, ah ; join 6 and e ; then h e and e g will define 
the boundary of shadow. 




Fig. 848. 

505. — To find the shadow cast hy an inclined shelf. In Fig. 
343, the plan and elevation of such a shelf is shown, having also 
one end wider than the other. Proceed as directed for finding 
the shadows of Fig. 341, and find the points, d and c ; then a d 
anl c? c win be the shadow required. If the shelf had been 



SHADOWS. 



387 



parallel in width on the plan, then the Hne, d c, would have been 
parallel with the shelf, a b. 



e c 

LJ — 



1 

1 




1 

f\ 


Sill 




e c 



Fig. 344 Fig. 845. 

506. — To find the shadow cast by a shelf inclined in its ver- 
tical section either upward or downward. From a, {Fig. 3M 
and 345,) draw a 6, at the usual angle, and from 6, draw b c, 
parallel with the shelf; obtain the point, e, by drawing a line 
from J, at the usual angle. In Fig. 344, join e and i ; then i e 
and e c will define the shadow. In Fig. 345, from o, draw o i, 
parallel with the shelf; join i and e ; then i e and e c will be the 
shadow required. 

The projections in these several examples are bounded by 
straight lines ; but the shadows of curved lines may be found in 
the same manner, by projecting shadows from several points in 
the curved hue, and tracing the curve of shadow through these 
points. Thus — 





Kg. 847. 



ng, 846. 



388 



AMERICAN HOTjSE-CARPENTER. 



507. — To find the shadow of a shelf having its front edge, or 
end, curved on the plan. In Fig. 346 and 347, A and A show an 
example of each kind. From several points, as a, a, in the plan, 
and from the corresponding points, o, o, in the elevation, draw 
rays and perpendiculars intersecting at e, e, &c. ; through these 
points of intersection trace the curve, and it will define the shadow. 



N 


e 


UllUiiiiiiiiiiiiiiii 1 


iiitiiiiiiiiiii.iiiiiiiiiiiiiiiiii 




/ 





Fig. 848. 

608. — To find the shadow of a shelf curved in the elevation. 
In Fig. 348, find the points of intersection, e, e and e, as in the 
last examples, and a curve traced through them will define the 
shadow. 

The preceding examples show how to find shadows when cast 
upon a vertical plane ; shadows thrown upon curved surfaces are 
ascertained in a similar manner. Thus — 




SHADOWS. 



389 



509. — To find the shadow cast upon a cylindrical wall by a 
vrojection of any kind. By an inspection of Fig. 349, it will be 
seen that the only difference between this and the last examples, 
is, that the rays in the plan die against the circle, a b, instead of 
a straight line. 




Fig. 850. 

510. — To find the shadow cast by a shelf upon an inclined 
vmll. Cast the ray, a b, {Fig. 350,) from the end of the shelf to 
the face of the wall, and from b, draw b c, parallel to the shelf ; 
cast the ray, d e, from the end of the shelf; then the lines, d e 
and € c, will define the shadow. 

These examples might be multiplied, but enough has been 
given to illustrate the general principle, by which shadows in all 
instances are found. Let us attend now to the application of this 
principle to such famihar objects as are likely to occur in practice, 




Fig. 851. 



a90 



AMERICAN HOUSE-CARPENTER. 



511. — To find the shadow of a projecting' horizontal beam 
From trie points, a, a, &c., {Fig. 351,) cast rays upon the wall . 
the intersections, e, e, e, of those rays with the perpendiculars 
drawn from the plan, will define the shadow. If the beain be in- 
clined, either on the plan or elevation, at any angle other than a 
light angle, the difference in the manner of proceeding can be seen 
by reference to the preceding examples of inclined shelves &c. 




Fig. 852. 

512. — To find the shadow in a recess. From th< point, a, 
[Fig. 352,) in the plan, and h in the elevation, draw th rays, a c 
and b e ; from c, erect the perpendicular, c e, and frua e, draw 
the horizontal line, e d ; then the lines, c e and e d, will show the 
extent of the shadow. This applies only where the back of tlie 
recess is parallel with the face of the wall. 




Fig. 353. 



513. — To find the shadow in a recess, when the face of the 
wall is inclined, and the back of the recess is vertical. In Fig. 
353, A shows the section and B the eler ation of a recess of this 



SHADOWS. 



391 



kind. From h, and from any other point in the hne, ho as a 
draw the rays, h c and a e ; from c, a, and e, draw the hcrizonta'. 
Unes, eg, a f, and eh ; from d and /, cast the rays, d i and / h • 
from /, through A, draw i 5 ; then s i and i g will define thfc 
shadow. 

d 




Fig. 854. 



514. — To find the shadow in a fireplace. From a and h, 
{Fig. 354,) cast the rays, a c and 6 e, and from c, erect the per- 
pendicular, c e ; from e, draw the horizontal line, e 0, and join ( 
and cZ ; then €6,6 0, and o cf, will give the extent of the shadow. 




|i|ii|iiiiiiiiiiiJ iiii|ii[[iiMliniiHf 



Fig. 855. 



515. — To find the shadow of a moulded window-lintel. Casl 
rays Irom the projections, a, c, & :., in the plan, {Fig, 355,) and 
d, e, &c., in the elevation, and draw the usual perpendiculars in- 
tersecting the rays at 2, i, and i ; these intersections connected 



AMERICAI^ HOUSE-CARPENTER. 



and horizontal lines drawn franr them, will define the shadow 
The shadow on the face of the lintel is found by casting a ra"y 
back from i to s, and drawing the horizontal line, s n. 




Fig. 856. 



516. — To find the shadow cast hy the nosing of a step. Fiom 
«, {Fig. 356,) and its corresponding point, c, cast the rays, a L 
and c d, and from b, erect the perpendicular, b d ; tangical to the 
curve at e, cast the ray, e f, and from e, drop the perpendicular, 
e o, meeting the mitre-line, a gy in o ; cast a ray from o to i, and 
from I, erect the perpendicular, i f ; from h, draw the ray, hk; 
from f to d and from d to k, trace the curve as shown in the 
figure ; from k and h, draw the horizontal Hnes, k n and h s ; then 
the limit of the shadow will be completed. 

517. — To find the shadow thrown by a pedestal upon steps. 
From o, [Fig. 357,) in the plan, and from c in the elevation, draw 
the rays, a b and c e ; then a o will show the extent of the shadow 
on the first riser, as at ^ ; f g will determine the shadow on the 
" second riser, as at jB ; c d gives the amount of shadow on the 
first tread, as at C, and h i that on the second tread, as at D ; 
which completes the shadow of the left-hand pedestal, both on the 
plar and elevation. A mere inspection of the figure will be suf- 



SHADOWS. 



393 





Fig. 857. 

ficient to show how the shadow of the right-hand pedestal is 
obtained. 





Fig. 358. Fig 

518. — To find the shadow thrown on a column hy a square 
abacus. From a and 6, {Fig. 358,) draw the rays, a c and h e, 
and from c, erect the perpendicular, c e ; tangical to the curve at 
d, draw the ray, d fi and from h, corresponding to / in the plan, 
draw the ray, ho; take any point between a and/, as z, and from 
this, as also from a corresponding point, n, draw the rays, i r and 
n s ; from r, and from d^ erect the perpendiculars, r s and do; 
through the points, e, s, and o, trace the curve as shown in the 
figure ; then the extent of the shadow vrill be defined. 

519. — To find the shadow thrown on a column hy a circular 
abacus. This is so near like the last example, that no explanation 
will be necessary farther than a reference to the preceding article 

50 



394 



AMERICAN HOUSE-CARPENTEK, 




Fig. 360. 



520. — To find the shadows on the capital of a column. This 
may be done according to the principles explained in the examples 
already given ; a quicker way if doing it, however, is as follows 
If we take into consideration one ray of light in connection with 
all those perpendicularly under and over it, it is evident that these 
several rays would form a vertical plane, standing at an angle of 
45 degrees with the face of the elevation. Now, we may sup 
pose the column to be sliced, so to speak, with planes of tliis 



SHADOWS. 



395 









III 
1 1 1 1 


iriiiiiiii'" 

Mi 

Mill II 1 1 1 M : 1 




Fig. 861. 



nature — cutting it in the lines, ah, c d^ &c., {Fig. 360,) and, in 
the elevation, find, by squaring up from the plan, the lines of sec- 
tion which these planes would make thereupon. For instance : 
in finding upon the elevation the line of section, a b, the plane 
cuts the ovolo at e, and therefore /will be the corresponding point 
upon the elevation ; h corresponds with g, i with j, o with s, and 
I with h. Now, to find the shadows upon this line of section, cast 
from nif the ray, m n, from h, the ray, h o, &c. ; then that part ©f 
the section indicated by the letters, m f i n, and that part also be- 
tween h and o, will be under shadow. By an inspection of the 
figure, it will be seen that the same process is applied to each line 
of section, and in that way the points, p, r, t, u, v, to, x, as alsc 
1, 2, 3, (fee, are successively found, and the lines of shadow 
traced through them. 

Fig. 361 is an example of the same capital with all the shadows 
finished in accordance with the lines obtained on Fig. 360. 

521. — To find the shadow thrown on a vertical wall by a 
column and entablature standing n advance of said wall. Cast 



396 



AMERICAN HOUSE-CARPENTER 




Fig. 862. 



rays from a and &, {Fig. 362,) and find the point, c, as in tlie 
previous examples ; from d^ draw the ray, d e, and from e, the 
horizontal hne, e f ; tangical to the curve at g and A, draw the 
rays, g j and h i, and from i and j, erect the perpendiculars, i I 
and j k ; from tti and n, draw the rays, m f and w A;, and trace the 
curve between k and /; cast a ray from o to p, a vertical line 
from p to 5, and through 5, draw the horizontal liae, s t ; the 
shadow as required will then be completed. 



SHADOWS. 



397 




Fig. 863. 



Fig. 363 is an example of the same kind as the last, with all 
the shadows filled in, according to the lines obtained in the pre- 
ceding figure. 




Fig. 864. 



522 — Fig, 364 and 365 are examples of the Tuscan cornice. 
The manner of obtainirg the shadows is evident. 



398 



AMERICAN HOUSE-CARPENTER. 




Pig. 865. 

523. — Reflected light. In shading, the finish and hfe of an 
object depend much on reflected light This is seen to advantage 
in Fig. 361 and on the column in Fig. 363. Reflected rays are 
thtown in a direction exactly the reverse of direct rays ; therefore, 
on that part of an object v^^hich is subject to reflected light, the 
shadow^s are reversed. The fillet of the ovolo in Fig. 361 is an 
example of this. On the right-hand side of the column, the. face 
of the fillet is much darker than the cove directly under it. The 
reason of this is, the face of the fillet is deprived both of direct 
and reflected light, w^hereas the cove is subject to the latter. 
Other instances of the effect of reflected light wrill be seen in the 
other examples. 



APPENDIX. 



ALGEBKAICAL SIG:t^S. 



-f , plus^ signifies addition, and that the two quantities between which 
it stands are to be added together ; as a + 6, read a added to h. 

— , minus, signifies subtraction, or that of the two quantities between 
which it occurs, the latter is to be subtracted from the former ; as 
a — b, read a minus b. 

X , multiplied by, or the sign of multiplication. It denotes that the two 
quantities between which it occurs are to be multiplied together ; 
as a X i, read a multiplied by 6, or a times b. This sign is usually 
omitted between symbols or letters, and is then understood, as ab. 
This has the same meaning as a x ft. It is never omitted between 
arithmetical numbers ; as 9 x 5, read nine times five. 

■4- , divided by, or the sign of division, and denotes that of the two quanti- 
ties between which it occurs, the former is to be divided by the latter ; 
as a -f- 6, read a divided by b. Division is also represented thus : 

-) in the form of a fraction. This signifies that a is to be divided by 

^ b. When more than one symbol occurs above or below the line, 

or both, as , it denotes that the product of the symbols above 

the line is to be divided by the product of those below the line. 

=, is equal to, or sign of equality, and denotes that the quantity or 
quantities on its left are equal to those on its right ; as a -— 6 =: cr, 
read a minus b is equal to c, or equals c ; or, 9 — 5 = 4, read nine 
minus five equals four. This sign, together with the symbols on 
each side of it, when spoken of as a whole, is called an equation. 

a^ denotes a squared, or a multiplied by a, or the second power of a, 
and 

o} denotes a cubed, or a multiplied by a and again multiplied by a, or 
the third power of a. The small figure, 2, 3, or 4, &c., is termed 
the index or exponent of the power. It indicates how many times 
the symbol is to be taken. Thus, a^ =. aa, a^ =aaa, a* =z aa aa. 

"v/ is the radical sign, and denotes that the square root of the quantity 
following it is to be extracted, and 

51 



4 APPENDIX. 

y/ denotes that the cube root of. the quantity following it is to he ex- 
tracted. Thus, VS = 3, and v^2'7 = 3. The extraction of roots 
is also denoted by a fractional index or exponent, thus 

a^ denotes the square root of a, 

a^ denotes the cube root of a, 

a'^ denotes the cube root of the square of a, &c. 



TRIGONOMETEICAL TERMS. 




Pig. 868. 



In Fig. 366, where AB \% the radius of the circle B C H^ draw a 
line A F^ from A^ through any point, (7, of the arc B G. From C draw 
C D perpendicular to ^ ^ ; from B draw B E perpendicular to AB ; 
and from G draw G F perpendicular to A G. 

Then, for the angle FAB, when the radius A C equals unity, CD 
is the sine ; AD the cosine ; D B the versed sine ; B E the tangent , 
G F the cotangent ; AE the secant ; and A F the cosecant. 



6 



APPENDIX. 



But if tlie angle be larger than one right angle, yet less than twc 
right angles, &&B A R, extend If A to K and U£ to IT, and from J5 
draw IT J perpendicular to A J. 

Then, for the angle BAIT, when the radius A IT equals unity, JIJ is 
the sine ;AJ the cosine ; BJ the versed sine ; B K the tangent ; and 
A K the secant. 

When the number of degrees contained in a given angle is known, 
then the value of the sine^ cosine^ dc, corresponding to that angle, may 
be found in a table of Natural Sines, Cosines, (fee. 

In the absence of such a table, and when the degrees contained in the 
given angle are unknown, the values of the sine, cosine, &c., may 
be found by computation, as follows: — Let B A C, {Fig. 367,) be the 




Fig. 867. 



given angle. At any distance from A^ draw c perpendicular to ^ ^. 
By any scale of equal parts obtain the length of each of the three lines 
a, 5, c. Then for the angle at A we have, by proportion, 

c 

a : c :: 1-0 : sm. = -. 



1-0 



1-0 



1-0 



1-0 



COS. 



tan. = 



cot. = 



sec = 



1-0 : cosec. 



a 

b 

a 

c 

V 

h 

c' 

a 

b' 

a 



Or, in any right angled triangle, for the angle contained between the 
base and hypothenuse — 

Wher perp. divided by hyp., the quotient equals the sine. 
« base " " hyp., " " " " cosine. 

« perp. " " base, " " " " tangent. 

" base " " perp., " " " " cotangent. 

« hyp. « " base, " 



%P- 



perp. 



cosecant. 



GLOSSARY 



Terms not found here can be found in the lists of definitions In other parts of *.his book, 
or in common dictionaries. 



— Abacus. — The uppermost member of a capital. 

Abhatoir. — A slaughter-house. 

Abbey. — The residence of an abbot or abbess. 
— Abutment. — That part of a pier from which the arch springs. 

Acanthus. — A plant called in English, beards-breech. Its leaves are 
employed for decorating the Corinthian and the Composite capitals. 
^ Acropolis. — The highest part of a city ; generally the citadel. 

Acroteria. — The small pedestals placed on the extremities and apex 
oi" a pediment, originally intended as a base for sculpture. 

Aisle. — Passage to and from the pev/s of a church. In Gothic ar- 
chitecture, the lean-to wings on the sides of the nave. 

Alcove. — Part of a chamber separated by an esirade, or partition of 
columns. Recess with seats, &c., in ggrdens. 

Altar. — A pedestal whereon sacrifice was offered. In modern 
churches, the area within the railing in front of the pulpit. 

Alto-relievo. — High relief; sculpture projecting from a surface so as 
to appear nearly isolated. 

Amphitheatre —fh double theatre, employed by the ancients for the 
exhibition of gladiatorial fights and other shows. 

Ancones. — Trusses employed as an apparent support to a cornice 
upon the flanks o^' the architrave. 

Annulet. — A small square moulding used to separate others ; the 
filLets in the Doric capital under the ovolo, and those which separate 
the flutings of columns, are known by this term. 

AntcB. — A pilaster attached to a wall. 

Apiary. — A place for keeping beehives. 

Arabesque. — A building after the Arabian style. 
— - Areostyle. — An intercolumniation of from four to five diameters. 

Arcade — A series of arches. 

Arch. — An arrangement of stones or other material in a curvilinear 
form, so as to perform the office of a lintel and carry superincumbent 
weights. 

Architrave. — That part of the entablature whi )h rests upon the 
capital of a column, and is beneath the frieze. The casing and 
mouldings about a door or window. 



APPENDIX. 

^ Archivolt. — The ceiling of a vault ; the under surface of an aich. 

Area. — Superficial measurement. An open space, below the leve 
of the ground, in front of basement windows. 

Arsenal. — A public establishment for the deposition of arms and 
warlike stores. 

Astragal. — A Bmall moulding consisting of a half-round with a fillet 

on each side. 

Attic. — A low story erected over an order of architecture. A low 
additional story immediately under the roof of a building. 

Aviary. — A place for keeping and breeding birds. 

Balcony. — An open gallery projecting from the front of a building. 

\ ^ Baluster. — A small pillar or pilaster supporting a rail. 

- Balustrade. — A series of balusters connected by a rail. 
, Barge-course. — That part of the covering which projects over the 
gable of a building. 

Base. — The lowest part of a wall, column, &c. 
Basement-story. — That which is immediately under the principal 
story, and included within the foundation of the building. 

Basso-relievo. — Low relief ; sculptured figures projecting from a 
surface one-half their thickness or less. See Alto-relievo. 
I Battering. — See Talus. 
Battlement. — Indentations on the top of a wall or parapet. 
Bay-window. — A window projecting in two or more planes, and not 
forming the segment of a circle. 

Bazaar. — A species of mart or exchange for the sale of various ar- 
tides of merchandise. 
V,, Bead. — A circular moulding. 
V Bed-mouldings. — Those mouldings which are between the corona 
and the frieze. 

Belfry. — That part of a steeple in which the bells are hung : an- 
ciently called campanile.^ 

Belvedere.— An ornamental turret or observatory commanding a 
pleasant prospect. 

\. BoiD-window. — A window projecting in curved lines. 
^^ Bressummer. — Abeam or iron tie supporting a wall over a gateway 
or other opening. 

Brick-nogging. — The brickwork between studs of partitions. 
\^ Buttress. — A projection from a wall to give additional strength. 

,. Cable. — A cylindrical moulding placed in flutes at the lower part of 

the column. 

v/ Camber. — To give a convexity to the upper surface of a beam. 
\ Campanile. — A tower for the reception of bells, usually, in Italy, 

separated from the church. 

Canopy. — An ornamental covering over a seat of state. 
X^- Cantulivers. — The ends of rafters under a projecting roof. Pieces 
^ of wood or stone supporting the eaves. 

\ Capital. — The uppermost part of a column included between the 

shaft and the architrave. 



APPEND IX. y 

l^ Caravansera. — In the East, a large public building for the reception 
of travellers by caravans in the desert. 

Carpentry. — (From the Latin, carpentum, carved wood.) That de- 
partment of science and art which treats of the disposition, the con- 
struction and the relative strength of timber. Th? first is called de- 
scriptive, the second constructive, and the last mechanical carpentry. 

Caryatides. — Figures of women used instead of columns to support 
an entablature. 
y^ Casino. — A small country-house. 

Castellated. — Built with battlements and turrets in imitation of an- 
cient castles. 

Castle. — A building fortified for military defence. A house with 
towers, usually encompassed with walls and moats, and having a don- 
jon, or keep, in the centre. 

Catacombs. — Subterraneous places for burying the dead. 

Cathedral. — The principal church of a province or diocese, wherein 
the throne of the archbishop or bishop is placed. 
\^^ Cavetto. — A concave moulding comprising the quadrant of a circle. 

Cemetery. — An edifice or area where the dead are interred. 
Y^^Cenotaph. — A nionument erected to the memory of a person buried 
in another place. 

^^ Centring. — The temporary woodwork, or framing, whereon any 
vaulted work is constructed. 

Cesspool. — A well under a drain or pavement to receive the waste- 
water and sediment. 

Chamfer. — The bevilled edge of any thing originally right-angled. 
. Chancel. — That part of a Gothic church in which the altar is placed. 

Chantry. — A little chapel in ancient churches, with an endowment 
for one or more priests to say mass for the relief of souls out of purga- 
tory. 

Chapel. — A building for religious worship, erected separately from 
a church, and served by a chaplain. 

Chaplet. — A moulding carved into beads, olives, &;c. 
--' Cincture. — The ring, listel, or fillet, at the top and bottom of a co- 
lumn, which divides the shaft of the column from its capital and base. 

Ctrcus. — A straight, long, narrow building used by the Romans for 
the exhibition of public spectacles and chariot races. At the present 
day, a building enclosing an arena for the exhibition of feats of horse, 
manship. 
,^ Clerestory. — The upper part of the nave of a church above the 
roofs of the aisles. 

Cloister. — The square space attached to a regular monastery or 
large church, having a peristyle or ambulatory around it, covered with 
a range of buildings. 

Coffer-dam. — A case of piling, water-tight, fixed in the bed of a 
river, for the purpose of excluding the water while any work, such as 
a wharf, wall, or the pier of a bridge, is carried up. 

Collar-beam. — A horizontal beam fra med between two principal 
cafters above the tie-beam. 
^ tullmiade. — A range of columns. 

Columbarium. — A pigeon-house. 



10 



APPENDIX. 



■'Column. — A vertical, cylindrical support under the entablature ol 
an order. 

Common-rafters. — The same as jack-rafters, which see 

Conduit. — A long, narrow, walled passage underground, for secret 
^^ communication between different apartments. A canal or pipe for the 
conveyance of water. 

Conservatory. — A building for preserving curious and rare exotic 
plants. 
u-^ Consoles. — The same as ancones, which see. 

Contour. — The external lines which bound and terminate a figure. 

Convent. — A building for the reception of a society of religious per- 
sons. 
_^. Coping. — Stones laid on the top of a wall to defend it from the 
weather. 

Corbels. — Stones or timbers fixed in a wall to sustain the timbers of 
'a floor or roof. 
.- Cornice. — Any moulded projection which crowns or finishes the 
part to which it is affixed. 

Corona. — That part of a cornice which is between the crown- 
mouldincr and the bed-mouldinsfs. 

Cornucopia. — The horn of plenty. 

Corridor. — An open gallery or communication to the different apart- 
ments of a house. 

Cove. — A concave moulding. 

Cripple-rafters. — The short rafters which are spiked to the hip-rafter 
of a roof. 

Crockets. — In Gothic architecture, the ornaments placed along the 
angles of pediments, pinnacles, &c. 
. Crosettes. — The same as ancones, which see. 

Crypt. — The under or hidden part of a building 

Culvert. — An arched channel of masonry or brickwork, built be- 
neath the bed of a canal for the purpose of conducting v/ater under it. 
\ny arched channel for water underground. 

V Cupola. — A small building on the top of a dome. 

y^ Curtail -step. — A step with a spiral end, usually the first of the flight. 
Cusps. — The pendents of a pointed arch. 

V Cyma. — An ogee. There are two kinds ; the cyma-recia, having 
the upper part concave and the lower convex, and the cyma-reversa, 
with the upper part convex and the lower concave. 

^^Dado. — The die, or part between the base and cornice of a pedestal. 
Dairy. — An apartment or building for the preservation of m.ilk, and 
the manufacture of it into butter, cheese, &c. 

^ Dead-shoar. — A piece of timber or stone stood vertically in brick- 
work, to support a superincumbent weight until the brickwork v/hich 
is to carry it has sei or become hard. 

Decastyle. — A building having ten columns in front. 
Dentils. — (From the Latin, denies, teeth.) Small rectangular blocks 
^ . used in the bed-mouldings of some of the orders. 

Diastyie. — An iniercolumniation of three, or, as some say, foui" 
diameters. 



APPENDIX. 



n 



- Vie. — That part of a pedestal included between the lase and the 

cornice ; it is also called a dado. 

,, Dodecastyle. — A building having twelve columns in front. 

Donjon. — A massive tower within ancient castles to which the gar- 
-ison might retreat in case of necessity. 
vDooks. — A Scotch term given to wood?n bricks. 

Dormer. — A window placed on the roof of a house, the frame being 
p.aced vertically on the rafters. 

Dormitory. — A sleeping-room. 

Dovecote. — A building for keeping tame pigeons. A columbarium. 

-yEchinus. — The Grecian ovolo. 

Elevation. — A geometrical projection drawn on a plane at right an- 
gles to the horizon. 

y Entablature. — That part of an order which is supported by the co- 
lumns ; consisting of the architrave, frieze, and cornice. 
■yEustyle. — An intercolumniation of two and a quarter diameters. 

Exchange. — A building in which merchants and brokers meet to 
transact business. 
. Extrados. — The exterior curve of an arch. 

<?' Facade. — The principal front of any building. 
y Face-mould — The pattern for marking the plank, out of which hand- 
railing is to be cut for stairs, &c. 

^,, Facia, or Fascia. — A flat member like a band or broad fillet. 
\ ^.Falling-mould. — The mould applied to the convex, vertical surface 
of the rail-piece, in order to form the back and under surface of the 
rail, and finish the squaring. 

Festoon. — An ornament representing a wreath of flowers and leaves. 
Fillet. — A narrow flat band, listel, or annulet, used for the separa- 
cion of one moulding from another, and to give breadth and firmness 
to the edges of mouldings. 

Flutes. — Upright channels on the shafts of columns. 
Flyers. — Steps in a flight of stairs that are parallel to each other. 
Forura. — >In ancient architecture, a public market ; also, a place 
where the common courts were held, and law pleadings carried on. 

Foundry. — A building in which various metals are cast into mould? 
or shapes. 
^ Frieze. — That part of an entablature included between the archi- 
trave and the cornice. 

/ Gable. — The vertical, triangular piece of wall at the end of a root, 

from the level of the eaves to the summit. 

y Gain. — A recess made to receive a t-'non or tusk. 

Gallery. — A common passage to several rooms in an upper storv. 

A long room for the reception of pictures. A platform raised on co- 

lumnr, pilasters, or piers. 
\^- Girder. — The principal beam in a floor for supporting the binding 
""and other joists, whereby the bearing or length is lessened, 
v^ Glyph. — A vertical, sunken channel. From their number, those in 
"the Doric order are called triglyphs. 



12 APPENDIX. 

Granary. — A building for storing grain, especially that intended to 
be kept for a considerable time. 

Groin. — The line formed by the intersection of two arches, which 
cross each other at any angle. 

GuttcB. — The small cylindrical pendent ornaments, otherwise called 
drops, used in the Doric order under the triglyphs, and also pendent 
from the mutuli of the cornice. 

Gymnasium. — Orig" oally, a space measured out and covered with 
sand for the exercise ( f athletic games *. afterwards, spacious buildings 
devoted to the mental as well as corporeal instruction of youth. 

Hall. — The first large apartment on entering a house. The public 
room of a corporate body. A manor-house. 

Ham. — A house or dwelling-place. A street or village : hence 
Notting/ia7/i, Bucking/iam, &c. Hamlet, the diminutive of ham, is a 
small street er village. 

Helix. — The small volute, or twist, under the abacus in the Corin- 
thian capital. 
. Hem. — The projecting spiral fillet of the Ionic capital. 

Hexastyle. — A building having six columns in front. 

Hip-rafter. — A piece of timber placed at the angle made by two ad- 
jacent inclined roofs. 

Homestall. — A mansion-house, or seat in the country. 

Hotel, or Hostel. — A large inn or place of public entertainment. A 
large house or palace. 

Hot-house. — A glass building used in gardening. 

Hovel. — An open shed. 

Hut. — A small cottage or hovel generally constructed of earthy 
materials, as strong loamy clay, &c. 

..^ Impost. — The capital of a pier or pilaster which supports an arch. 

Intaglio. — Sculpture in which the subject is hollowed out, so that 
the impression from it presents the appearance of a bas-relief 

Intercolumniation. — The distance between two columns. 
\^ Intrados. — The interior and lower curve of an arch. 

Jack-rafters. — Rafters that fill in between the principal rafters of a 
roof; called also common-rafters. 

Jail. — A place of legal confinement. 
\y Jamhs. — The vertical sides of an aperture. 
V .Joggle-piece. — A post to receive struts. 

Joists. — The timbers to which the boards of a floor or the laths of a 
ceiling are nailed. 

Keep. — The same as donjon, which see. 
Key-stone. — The highest central stone of an arch. 
Kiln. — A building for the accumulation and retention of heat, in Ol- 
der to dry or burn certain materials deposited within it. 
King-post. — The centre-post in a trussed roof 
Knee, — A convex bend in the back of a hand-rail. See Ramp. 



APPENDIX. 13 

^Lactarium. — The same as dairy, which see. 
/ Lantern. — A cupola having windows in the sides for lighting an 
apartment beneath. 
,.- Larmier. — The same as corona, which see. 

Lattice. — A reticulated window for the admission of air, rather than 
light, as in dairies and cellars. 
,^. Lever-hoards. — Blind-slats : a set of boards so fastened that they 
may be turned at any angle to admit more or less light, or to lap upon 
3ach other so as to exclude all air or light through apertures. 

LinteL—^A piece of timber or stone placed horizontally over a door, 
window, or other opening. 

Listel. — The same as fillet, which see. 

Lobby. — An enclosed space, or passage, communicating with the 
principal room or rooms of a house. 

Lodge. — A small house near and subordinate to the mansion. A 
cottage placed at the gate of the road leading to a mansion. 

Loop. — A small narrow window. Loophole is a term applied to the 
vertical series of doors in a warehouse, through which goods are de- 
livered by means of a crane. 

•^^^huffer -boar ding. — The same as lever-boards, which see. 
\... Luthern. — The same as dormer, which see. 

--- Mausoleum, — A sepulchral building — so called from a very cele- 
brated one erected to the memory of Mausolus, king of Caria, by his 
wife Artemisia. 
^ Metopa. — The square space in the frieze between the triglyphs of 
the Doric order. 
' Mezzanine. — A story of small height introduced between two of 
greater height. 

^' Minaret. — A slender, lofty turret having projecting balconies, com- 
mon in Mohammedan countries. 

Minster. — A church to which an ecclesiastical fraternity has been 
or is attached. 

Moat. — An excavated reservoir of water, surroundmg a house, cas 
• tie or town. 

■— Modillion. — A projection under the corona of the richer orders, re 
•embling a bracket. 

- Module. — The semi-diameter of a column, used by the architect as 
a measure by which to proportion the parts of an order. 

Monastery. — A building or buildings appropriated to the reception of 
monks. 

v^ Monopteron. — A circular collonade supporting a dome without an 
enclosing wall. 

Mosaic. — A mode of representing objects by the inlaying of small 
cubes of glass, stone, marble, shells, &;c. 

Mosque. — A Mohammedan temple, or place of worship. 

Mullions. — The upright posts or bars, which divide the lights in a 
Gothic window. 

^ Muniment-house. — A strong, fire-proof apartment for the keeping 
and preservation of evidences, charters, seals, &c., called muniments. 



14: APPENDIX, 

Mtcseum. — A repository of natural, scientific and literary ^ curvosities 
or of works of art. 
^ Mutule. — A projecting ornament of the Doric cornice supposed tc 
represent the ends of rafters. 

Nave. — The main body of a Gothic church. 

Newel. — A post at the starting o- landing of a flight of stairs. 

Niche. — A cavity or hollow place in a wall for the reception of a 
statue, vase, &c. 

iVb^5.— Wooden bricks. 

Nosing. — The rounded and projecting edge of a step in stairs. 

Nunnery. — A building or buildings appropriated for the reception of 
Quns. 

Obelisk. — A lofty pillar of a rectangular form. 
Octastyle.—k. building with eight columns in front. 
Odeum. — Among the Greeks, a species of theatre wherein the poets 
and musicians rehearsed their compositions previous to the public pro- 
duction of them. 
^Ogee. — See Cyma. 

V- Orangery. — A gallery or building in a garden or parterre fronting 
the south. 
^ Oriel-window. — A large bay or recessed window in a hall, chapel, or 

other apartment. 
^ Ovolo. — A convex projecting moulding whose profile is the quad- 
rant of a circle. 

Pagoda. — A temple or place of worship in India. 

Palisade. — K fence of pales or stakes driven into the ground. 
y Parapet.- — A small wall of any material for protection on the sides 
of bridges, quays, or high buildings. 

Pavilion. — A turret or small building generally insulated and com- 
'"prised under a single roof. 

Pedestal. — A square foundation used to elevate and sustain a co- 
lumn, statue, &c. 

Pediment. — The triangular crowning part of a portico or aperture 
"•which terminates vertically the sloping parts of the roof: this, in 
Gothic architecture, is called a gable. 

Penitentiary. — A prison for the confinement of criminals whose 
crimes are not of a very heinous nature. 

Piazza. — A square, open space surrounded by buildings. This 
term is often improperly used to denote a portico. 

Pier. — A rectangular pillar without any regular base or capital. 
The upright, narrow portions of walls between doors and windows are 
known by this term. 

Pilaster. — A square pillar, sometimes insulated, but more common 
ly engaged in a wall, and projecting only a part of its thickness. 

Piles. — Large timbers driven into the ground to make a secure 
foundation in marshy pla :5es, or in the bed of a river. 
/ Pillar. — A column of irregular form, always disengaged, and al- 



APPEND- X. 15 

ways deviating from the proportions ol the orders ; wlience the distinc- 
tion between a pillar and a column. 

^^ Pinnacle. — A small spire used to ornament Gothic buildings. 
■^ Planceer. — The same as soffit, which see. 

"^ Plinth. — The lower square member of the base of a column, pedes- 
tal, or wall. 

Porch. — An exterior appendage to a building, forming a covered 
appioach to one of its principal doorways. 

V Portal.— The aich over a door or gate ; the framework of the gate ; 
the lesser gate, when there are two of different dimensions at one en- 
trance. 

Portcullis. — A strong timber gate to old castles, made to slide up 
and down vertically. 

• Portico. — A colonnade supporting a shelter over a walk, or ambu- 
latory. 

Priory. — A building similar in its constitution to a monastery or 
abbey, the head whereof was called a prior or prioress. 

Prism. — A solid bounded on the sides by parallelograms, and on the 
ends by polygonal figures in parallel planes. 
■■^y Prostyle. — A building with columns in front only. 

, Purlines. — Those pieces of timber which lie under and at right an- 
gles to the rafters to prevent them from sinking. 
./ Pycnostyle. — An intercolumniation of one and a half diameters. 

Pyramid. — A solid body standing on a square, triangular or poly- 
gonal basis, and terminating in a point at the top. 

Quarry. — A place whence stones and slates are procured. 

Quay. — (Pronounced, key.) A bank formed towards the sea or on 
the side of a river for free passage, or for the purpose of unloading 
merchandise. 
\ Quoin. — An external angle. See Rustic quoins. 

' Rahhet, or Rebate. — A groove or channel in the edge of a board. 

, Ramp. — A concave bend in the back of a hand-rail. 

' Rampant arch. — One having abutments of different heights. 

^.Regula. — The band below the taenia in the Doric order. 

>, Riser. -^\x\ stairs, the v?rtical board forming the front of a step. 
. Rostrmn. — An elevated platform from which a speaker addresses an 
audience. 

V Rotunda. — A circular building. 

' RiibUe-iuall.-^—K wall built of unhewn stone. 
^ Rudenture. — The same as cahle, which see. 

Rustic quoins. — The stones placed on the external angle of a build- 
ing, projecting beyond the face of the wall, and liaving their edges 
bevilled. 

Rustic-work. — A mode of building masonry wherein the faces of the 
stones are left rough, the sides only being wrought smooth where the 
union of the stones takes place. 



16 APPENDIX. 

Salon, or Saloon. — A lofty and spacious apartment comprehending 
the height of two stories with two tiers of windows. 

Sarcophagus. — A tomb or coffin made of one stone. 

Scantling. — The measu.'e to which a piece of timber is to be or has 
been cut. 

Scarfing. — The joining of two pieces of timber by bolting or nailing 
transversely together, so that the two appear but one. 

Scotia. — The hollow moulding in the base of a column, between the 
fillets of the tori. 

Scroll. — A carved curvilinear ornament, somewhat resembling in 
profile the turnings of a ram's horn. 

Sepulchre. — A grave, tomb, or place of interment. 

Sewer. — A drain or conduit for carrying off soil or water from any 
place. 

Shaft. — The cylindrical part between the base and the capital of a 
column. 

Shoar. — A piece of timber placed in an oblique direction to support 
a building or wall. 
v/ Sill. — The horizontal piece of timber at the bottom of framing ; the 
timber or stone at the bottom of doors and windows. 

Sojit — The underside of an architrave, corona, &c. The underside 
of the heads of doors, windows, &;c. 

Summer. — The lintel of a door or window ; a beam tenoned into a 
girder to support the ends of joists on both sides of it. 
_. Systyle. — An intercolumniation of two diameters. 

T(Bnia. — The fillet which separates the Doric frieze from the archi- 
trave. 

Talus. — The slope or inclination of a wall, among workmen called 
battering. 

Terrace. — An area raised before a building, above the level of the 
ground, to serve as a walk. 

Tesselated pavement. — A curious pavement of Mosaic work, com- 
posed of small square stones. 

Tetrastyle. — A building having four columns in front. 

Thatch. — A covering of straw or reeds used on the roofs of cottages, 
barns, &c. 

Theatre. — A building appropriated to the representation of drama.. c 
spectacles. 

Tile. — A thin piece or plate of baked clay or other material used for 
the external covering of a roof. 

Tomb. — A grave, or place for the interment of a human body, in- 
cluding also any commemorative monument raised over such a place. 

Torus. — A moulding of semi-circular profile used in the bases of 
columns. 

Tower. — A lofty building of several stories, round or polygonal. 

Transept. — The transverse portion 'f a cruciform church. 

Transom. — The beam across a double-lighted window ; if the win 
dow have no transom, it is called a clerestory window. 



APPENDIX. 



17 



Tread. — That part of a step which is included between the face of 
its riser and that of the riser above. 

Trellis. — A reticulated framing made of thin bars of wood for 
screenr,, windows, &c. 

Triglyph. — The vertical tablets in the Doric frieze, chamfered on 
•.he two vertical edges, and having two channels in the middle. 

Tripod.- -A table or seat with three legs. 

Trochilus.— -The same as scotia, which see. 

Truss. — An arrangement of timbers for increasing the resistance to 
cross-strains, consisting of a tie, two struts and a suspending-piece. 

Turret. — A small tower, often crowning the angle of a wall, &c. 
V Tusk — A short projection under a tenon to increase its strength. 

Tympanum. — The naked face of a pediment, included between the 
•evel and the raking mouldings. 

^ ' Underpinning. — The wall under the ground-sills of a building. 

University. — An assemblage of colleges under the supervision of a 
senate, &c. 

Vault. — A concave arched ceiling resting upon two opposite paral- 
lel walls. 

Venetian- door. — A door having side-lights. 

Venetian- window. — A window having three separate apertures. 

Veranda. — An awning. An open portico under the extended roof 
of a building. 

Vestibule. — An apartment which serves as the medium of commu- 
nication to another room or series of rooms. 

Vestry. — An apartment in a church, or attached to it, for the pre- 
servation of the sacred vestments and utensils. 

Villa. — A country-house for the residence of an opulent person. 

Vinery. — A house for the cultivation of vines. 

Volute. — A spiral scroll, which forms the principal feature of the 
Ionic and the Composite capitals. 

Voussoirs. — Arch-stones 

\^ Wainscoting. — Wooden lining of walls, generally in panels. 

^.- Water-table. — The stone covering to the projecting foundation or 

other walls of a building. 

Well. — The space occupied by a flight of stairs. The space left 
beyond the ends of the steps is called the well-hole. 

Wicket. — A small door made in a gate. 
^ Winders. — In stairs, steps not parallel to each other. 

Zophorus. — The same as frieze, which see. 
^/ Zy^tos. — Among the ancients, a portico of unusual lei gth, common- 
ly appropriated to gymnastic exercises. 



TABLE OF SaUARES, CUBES, AND ROOTS. 

(From Hutton's Mathematics.) 



Na 
1 


Square. 


Cube. 


Sq. Root. 


CubeRoot. 


No. 


Square. 


Cube. 


Sq. Root. 


CubeRoot. 


1 


1 


1-0000000 


1000000 


68 


4624 


314432 


8-2462113 


4081655 


2 


4 


8 


1-4142136 


1-259921 


69 


4761 


328509 


8-3066239 


4 101586 


3 


9 


27 


1-7320508 


1-442250 


70 


4900 


343000 


8-3666003 


4-121285 


4 


16 


64 


2-0000000 


1-537401 


71 


5041 


357911 


8-4261498 


4-140818 


5 


25 


125 


2-2360680 


1-709976 


72 


5184 


373248 


8-4852814 


4-160168 


6 


36 


216 


2-4494897 


1-817121 


73 


53-29 


389017 


8-5440037 


4-179339 


7 


49 


343 


2-6457513 


1-912931 


74 


5476 


405224 


8-6023253 


4-198336 


8 


64 


512 


2-8284271 


2-000000 


75 


5625 


421875 


8-6602540 


4 217163 


9 


81 


729 


3-0000000 


2-080084 


76 


5776 


433976 


8-7177979 


4 235824 


10 


100 


1000 


3-1622777 


2-154435 


77 


5929 


456533 


8-7749644 


4-254321 


11 


121 


1331 


3-3165243 


2-223980 


78 


6084 


474552 


8-8317609 


4-272659 


12 


144 


1728 


3-4641016 


2-289429 


79 


6241 


493039 


8-8881944 


4-290640 


13 


169 


2197 


3-6055513 


2351335 


80 


6400 


512000 


8-9442719 


4-308869 


14 


196 


2744 


3-7416574 


2-410142 


81 


6561 


531441 


9-0000000 


4-326'/49 


15 


225 


3375 


3-8729833 


2-466212 


82 


6724 


551368 


9-0553851 


4-344481 


16 


256 


4096 


4-0000000 


2-519842 


83 


6839 


571787 


9-1104336 


4-36-2071 


17 


289 


4913 


4-1231056 


2-571232 


84 


7056 


592704 


91651514 


4-379519 


18 


324 


583-?, 


4-2426407 


2-620741 


85 


7225 


614125 


9-2195445 


4-396830 


19 


361 


6859 


4-3583989 


2-663402 


86 


7396 


636056 


9-2736185 


4-414005 


20 


400 


8000 


4-4721360 


2-714418 


87 


7569 


658503 


9-3-273791 


4 -'131048 


21 


441 


9261 


4-5825757 


2-758924 


88 


7744 


681472 


9-3808315 


4-447960 


22 


484 


10648 


4-6904158 


2-802039 


89 


7921 


704969 


9-4339811 


4-46'4745 


23 


529 


12167 


4-7958315 


2-843367 


90 


8100 


729000 


9-4858330 


4-481405 


24 


576 


13324 


4-8989795 


2-884499 


91 


8281 


753571 


95393920 


4-497941 


25 


625 


15625 


5-0000000 


2-924018 


92 


8464 


778683 


9-5916630 


4 514357 


26 


676 


17576 


5-0990195 


2-962496 


93 


8649 


804357 


9-6436508 


4-530655 


27 


729 


19683 


5-1961524 


3000000 


94 


8836 


830584 


9-6953597 


4 546836 


28 


784 


21952 


52915028 


3036589 


95 


9025 


857375 


9-7467943 


4 552903 


29 


841 


24389 


5-3351648 


3072317 


93 


9216 


884736 


9-7979590 


4-578357 


30 


900 


27000 


5-4772255 


3107232 


97 


9409 


912673 


9-8488578 


4-594701 


31 


961 


29791 


5 5677644 


3141331 


98 


9604 


941192 


98994949 


4-010436 


32 


1024 


32768 


5-6568542, 3-174802 


99 


9801 


970299 


9-9498744 


4-626065 


33 


1089 


35937 


5-7445326 3-207534 


100 


10000 


1000000 


10-0000000 


4 641589 


34 


1156 


39304 


5-8309519 3239612 


101 


10201 


1030301 


10-0498756 


4-657009 


35 


1225 


42875 


5-9160798 3-271066 


102 


10404 


1061208 


10-09950491 4^372329 


36 


1296 


46656 


6-0000000 


33J1927 


103 


10609 


1092727 


10- 14889 1G' 4-687548 


37 


1369 


50653 


6-0327625 


3-33-2222 


104 


10816 


1124864 


10-1980390' 4-702669 


38 


1444 


54872 


6-1644140 


3-361975 


105 


11025 


1157625 


10-2469508' 4-717694 


39 


1521 


59319 


6-2449980 


3-391211 


106 


11235 


1191016 


10-29563011 4-732623 


40 


1600 


64000 


6-3245553 


3 419952 


107 


11449 


1225043 


10-344080'tj 4-747459 
10-3923048; 4762203 


41 


1681 


68921 


6-4031-242 


3448217 


108 


11664 


1-259712 


^2 


1764 


74088 


6-4807407 


3-476027 


109 


11881 


1295029 


10-4403065' 4-776856 


43 


1849 


79507 


6-5574385 


3-503398 


110 


12100 


1331000 


10-4880885 4-791420 


44 


1936 


85184 


6-6332496 


3533348 


111 


12321 


1367631 


10-53565381 4-805895 


45 


2025 


91125 


6-7082039 


3-556893 


112 


12544 


1404928 


10-5330052! 4-820284 


46 


2116 


97336 


6-7823300 


3-583048 


113 


12769 


1442897 


10-6301458| 4-834588 


47 


2209 


103323 


6-8556546 


3-608826 


114 


12996 


1481544 


10-67707<J3' 4-848808 


48 


2304 


110592 


6-9282032 


3-634241 


115 


13225 


1520875 


10 7-2380n3i 4-862944 


49 


2401 


117649 


7-0000000 


3-659306 


116 


13456 


1560896 


10-770329'J 


4-876999 


50 


2500 


125000 


7-0710678 


3-634031 


117 


13689 


1601613 


10-8166533 


4-890973 


51 


2601 


132651 


7-1414284 


3-708430 


118 


13924 


1643032 


10-8627805 


4-904868 


52 


2704 


140608 


7-2111026 


3-732511 


119 


14161 


1685159 


10-9037121, 


4-918685 


53 


2809 


148877 


7-2301099 


3-756286 


120 


14400 


1728000 


10-9544512 


4-932424 


54 


2916 


157464 


7-3484692 


3-779763 


121 


14641 


1771561 


ll-OOOOOOO 


4-946087 


55 


3025 


166375 


7-4161985 


3-302952 


122 


14884 


1815848 


11-0453610 


4-959676 


56 


3136 


175616 


7-4833148 


3-825862 


123 


15129 


1860867 


11-09053^35 


4-973190 


57 


3219 


185193 


7-5498344 


3-S43501 


124 


15376 


1906624 


11-1355287 


4-986631 


58 


3364 


195112 


7-6157731 


3-870877 


125 


15625 


1953125 


11-180339J 


5-000000 


59 


3481 


205379 


7-6311457 


3-892996 


126 


15876 


2000376 


11-2249723 


5-013298 


60 


3600 


216000 


7-7459667 


3-914858 


127 


16129 


2048383 


11-2694277 


5-0265-26 


61 


3721 


226981 


7-8102497 


3-936497 


128 


16384 


2097152 


11-3137085 


5-039684 


62 


3S44 


238328 


7-8740079 


3-957891 


129 


16641 


2146689 


11-3578167 


5-052774 


63 


3969 


250017 


7-9372539 


3-979057 


130 


16900 


2197000 


11-4017543 


5065797 


64 


40% 


262144 


80000000 


4-000000 


131 


17161 


2248091 


11-4455231 


5-078753 


65 


4225 


274625 


3-0622577 


4-020726 


132 


17424 


2299968 


11-4891253 


5-091643 


66 


4356 


2874'j6 


8-1240334 


4041240 


133 


17689 


2352637 


11-5325626 


5 104469 


67 


448ii 


300763 


8-185352S 


4-061548 


134 


17956 


2406 J 04 


]< -5758369 


5 117230 



APPENDIX. 



19 



No. 


Square. 


Cube. 


Sq. Root. 


CubeRoot. 


No. 


Square. 


Cube. 


Sq. Root. 


CubeRoot. 


135 


18225 


2-160375 


11-6189500 


5-129923 


202 


4U804 


8242408! 14-2126704 


5-8674&4 


136 


18496 


2515156 


11-6619033 


5 142563 


203 


41-209 


8365427 


14-2478068 


5-877131 


137 


18769 


2571353 


11-7046999 


5-155137 


204 


41616 


8489664 


14-2328569 


5-886765 


133 


39044 


26^31)72 


11-7473401 


5167649 


205 


4-2025 


8615125 


14-3178211 


5-896368 


139 


V9321 


2635619 


11-7398261 


5-180101 


206 


4243f. 


8741816 


1 14-3527001 


5-905941 


140 


1960'J 


2744000 


11-8321596 


5-192494 


207 


42849 


8869743 


14-3874946 


5-915432 


141 


19381 


2303221 


11 •87434-22 


5-2043-28 


203 


43264 


8998912 


11-42-22051 


5-924992 


142 


*2UI64 


2363283 


11-9163753 


5-217103 


209 


43581 


9129329 


14-4558323 


5-934473 


143 


21)449 


2924207 


1 1-95326 J7 


5-229.321 


210 


44100 


9-261000 


14-4913767 


5-943022 


144 


20736 


2985984 


12-0000000 


5-241483 


211 


44-521 


9393931 


14-5258390 


5-953342 


145 


21025 


3048625 


12-0415946 


5-253533 


212 


44944 


95-281231 14-56021981 5-95-2732| 


146 


21316 


3112136 


12-0830460 


5-265637 


213 


45369 


9663597 14-5945195! 5-97-2093| 


147 


21609 


3176523 


12-1243557 


5-277632 


214 


45796 


9300344! 14-6287333 


5-931424 


148 


21904 


3241792 


12-1655251 


5-239572 


215 


46225 


9933375 14-66237831 5-990726| 


149 


22201 


3307J49 


12-2065555 


5 3,) 1459 


1 216 


46656 


10077696 14-6969385 


6-000000 


150 


22500 


3375000 


12-2474487 


5-313293 


217 


47089 


10218313 14-7309199 


6-009245 


151 


22301 


34-12951 


12-233-2057 5-3-25074 


218 


47524 


10.350232! 14-7648231 


6-013462 


152 


23104 


3511803 


12-3233280 5-336303 


219 


47961 


10503459 


14-7986486 


6-027650 


153 


23409 


3531577 


12-3693169| 5-343431 


220 


48400 


10648000 


14-8323970 


6-036811 


154 


23716 


36522'")4 


12-40967361 5-360103 


221 


43i41 


10793861 


14-8660687 


6-045943 


155 


24025 


3723375 


12-449399b 5-371635 


222 


49234 


10941048 


14-8996644 
14-9331845 


6-055049 


156 


2433!i 


37%416 


12-4399960 5-333213 


223 


49729 


11039567 


6-064127 


157 


24619 


3859393 


12-52996411 5-3J4691 


224 


50176 


112394-24 


14-9656295 


6-073178 


153 


24961 


3944312 


12-56J805l! 5-406120 


225 


50625 


11390625 


15-0000000 


6-082202 


159 


25281 


4019579 


12-6,)95-202l 5-417501 


226 


51076 


11543176 


15-0332964 


6-091199 


160 


2r.600 


4096000 


12-6491106! 5-423335 


227 


51529 


1 1697083 


150565192 


6-100170 


16 i 


25921 


4173281 


12-6335775 5-4401-22 


223 


51984 


11852352 


15-0996639 


6-109115 


162 


26244 


4251528 


12-7279-221 5-451352 


229 


52441 


12008939 


15-13-27460 


6-118033 


163 


26569 


4330747 


12-7671453 5-462556 


230 


52900 


12167000 


15-1657509 


6-126926 


164 


26896 


4410944 


12-8062485 


5-473704:! 231 


53361 


123-25391 


15-1986342 


6-135792 


165 


27225 


4492125 


12-8452326 


5-434807!! 232 


53824 


12487168 


15-2315462 


6-144634 


16J 


5i7556 


4574296 


12-8340987 


5-495365 


233 


54-289 


12649337 


15-2643375 


6-153449 


167 


27839 


4657463 


12 9228480 


5-5J6S78 


234 


54755 


1-2812904 


15-2970535 


6-162240 


168 


28224 


4741632 


12-9614814 


5-517848 


235 


55225 


1-2977875 


15-3297097 


6-171005 


169 


28561 


4826809 


13-0000000 


5-528775 


235 


55596 


13144256 


15 3622915 


6-179747 


170 


28900 


4913000 


13-0334048 


5-539658 


237 


55169 


1331-2053 


15-3948043 


6-183463 


171 


29241 


5000211 


13-0766968 


5-550499 


238 


55644 


13481272 


15-4272486 


6197154 


172 


29^84 


5083448 


13-1143770 


5-561293 


239 


57121 


13651919 


15-4596-248 


6-205822 


1 173 


29929 


5177717 


13-1529464 


5-572055 


240 


57600 


13324000 


15-4919334 


6-214465 


[ 174 


30276 


5268024 


13-1909060 


5 532770 


241 


58081 


139.^7521 


15-5241747 


6-223084 


175 


30625 


5359375 


13-2287566 


5-593445 


242 


58564 


14172438 


15-5583492 


6-231630 


1 176 


30976 


5451776 


13-2664992 


5-604079 


243 


59049 


14348907 


15-5334573 


6-240251 


17T 


31329 


5545233 


13-3041347 


5-614672 


244 


59536 


14526784 


15-6-204994 


6-248300 


i:8 


31684 


5639752 


13-3416541 


5-625-226 


245 


60025 


147061-25 


15-6524753 


6-257325 


179 


32041 


5735339 


13-3790832 


5-635741 


246 


60516 


14336936 


15-6843371 


6-255327 


ISO 


3^400 


5832000 


13-4164079 


5-646216 


247 


61009 


15059223 


15-716-2335 


6-274305 


181 


32761 


5929741 


13-4536-240 


5-656553 


248 


61504 


15252992 


15-7480157 


6-282761 


182 


33121 


6028568 


13-4907376 


5-667051 


249 


6-2001 


15433249 


15-7797333 


6-291195 


183 


33439 


6128487 


13-5277493 


5-677411 


250 


62500 


15525000 


15-8113333 


6-299605 


181 


33336 


6229504 


13-5646600 


5-637734 


251 


63001 


15313251 


15-8429795 


6-307994 


;• 185 


34225 


6331625 


13-6014705 


5-693019 


252 


63504 


16003008 


15-8745079 


6-316360 


1 18'^ 


34596 


6434856 


13-6331817 


5-708267 


253 


64009 


16194-277 


15-9059737 


6-324704 


187 


34969 


6539203 


13-6747943 5-718479 


254 


64516 


16387064 


15-9373775 


6-3330-26 


183 


35344 


6644672 


13-7113092 5-728554 


255 


65025 


16531375 


15 9687194 


6-341326 


189 


35721 


6751269 


13-7477271 


5-738794 


256 


65536 


16777216 


16 0000000 


6-349604 


'l90 


36100 


6859000 


13-7840488 


5-748397 


257 


65049 


16974593 


16-0312195 


6357861 


191 


35481 


6967871 


13-8202750 


5-753965 


253 


66564 


17173512 


16-0623734 


6-366097 


192 


36864 


7077838 


13-8564065 


5-763998 


259 


67031 


17373979 


160934769 


6-374311 


193 


3721.9 


7189057 


13-3924440! 5-778996] 


260 


67600 


17576000 


16-1245155 


6-382504 


194 


37o36 


7301384 


13-9283383! 5-783960| 


261 


63121 


17779531 


161554944 


6-390676 


195 


33025 


7414875 


13-964-2400 


5-798390 


262 


6H644 


17984723 


15-1884141 


6-393323 


196 


33416 


7529536 


14-0000000 


5-808736 


263 


69169 


13191447 


16-2172747 


6-406953 


197 


3330i) 


7645373 


14-0356633 


5-818548 


264 


69696 


18399744 


16-2480763 


6-415069 


198 


39204 


7762392 


14-0712473 


5-8-23477 


265 


702-25 


ISGJ'je^b 


16-2738206 


6-4-2310^ 


19W 


39601 


7880599 


14-1067360 


5-833-272 


265 


70756 


18821096 


16-3095064 


6-4312-23 


2UC 


40000 


8000000 


14-1421355 


5 848035 


267 


7123;> 


19034163 


16-3401346 


6-439277 


201 


40101 


8120601 


14-1774469 


5-857766 


263 


71324 


19243832 


16-3707055 


6-447306 



53 



20 



APPENDIX 



No. Square. 



269 
270 

271 1 
272i 
273i 

2741 
275 
276 
277 
278 
279 
290 
231i 
2S-2| 
233 
284 
235 
286' 
287| 
2381 
289| 
290' 
291 
292 
293 
294 
295 
296 
297 
298 
299 

3oo; 

301! 
3021 
303' 
304; 
305 
3a6j 
307 
3081 
3091 
310 
311 
312 
313 
3U 
315 
31fi 
317 
318^ 
319; 
32o! 
321! 
322! 
323I 
324' 
325| 
326i 
327: 
1 323 
3291 
330 
331 
332' 
333 
33ii 
335! 



Cube. 



7^351 
72900 
73441 
73984 
74529 
75076 
75625 
76176 
76729 
77284 
77811 
78400 
78951 
79524 
80089 
80656 
81225 
81796 
82369 
82944 
835211 
84100! 
846811 
85264' 
85849; 
86136' 
870251 
876161 
85209! 
88304! 
89401 
90000 
90601 
91204! 
91809| 
92416; 
93025! 
93636! 
942491 
94864, 
954811 
961001 
96721 
97344 
97969 
985961 
99225! 
99856 
100489! 
1011241 
101761 
102400 
103041 
103684 
104329 
104976 
105625 
106276 
106929 
107584 
108241 
108900 
109551 
110224 
110839 
111556 
1 12225 



Sq. Root. CubeRoot. No. Square. 



19465109 

19683000 

19902511 

20123648 

20346417 

20570824 

20796875 

21024576 

21253933 

21484952 

21717639 

21952000 

22188941 

2242576^ 

22665187 

22906304 

23149125 

23393656 

23639903 

23887872 

24137569 

24389000 

246421711 

24897083! 

251'S3757 

25412184 

25672375 

25934336 

26198073 

26463592 

26730899 

27000000 

27270901 

2754360s! 

27818127! 

28094464! 

28372625! 

23652616! 

28934443 

29218112 

29503529 

29791000 

30080231 

30371328 

30664297 

30959144 

31255375 

31554496 

31855013 

32157432 

32461759 

32768000 

33076161 

33336248 

33698267 

34012224 

34323125 

34645976 

34965783' 

35237552' 

35611239! 

359.37000: 

362646911 

36594368! 

3692603/1 

37259704; 

37595375! 



16-4012195 
16-4316767 
16-4620776 
16-4924225 
16-52-27116 
16-55-29454 
16-5331240 
16-6132477 
16-6433170 
16-6733320 
16-7032931 
16-7332005 
16-7630546 
16-7923556 
16-82-26033 
16-85-22995 
16-8319430 
169115345 
16-9410743 
16-9705627 
17-0000000 
17-0293864 
17-0537221 
17-0880075 
17-117-2423 
17-1464232 
17-1755640 
17-2046505 
17-2336379 
17-2626765 
17-2916165 
17-3-2050S1 
17-3493516 
17-3731472 
17-4063952 
17-4355958 
17-4642492 
17-4928557 
17-5214155 
17-5499288 
175783953 
17-6063169 
17-6351921 
17-6635217 
17-6918060 
17-7200451 
17-7432393 
17-7763338 
17-8044933 
17-8325545 
17-8605711 
17-8335433 
17-9164729 
17-9443534 
17-97-22308 
18-0000000 
18-0277564 
180554701 
180831413 
181107703 
181333571 
18-1659021 
18-1934054 
18-2208672 
18-2482376 
18-2756569 
18-3030052 



6-455315! 
6-4633041 
6-471274! 
6-4792241 
6-487154 
6-495065 
6-502957 
6-510833 
6-5186341 
6-526519 
6-534335? 
6-5421331 
6-549912 
6-557672 
6-555414! 
6-573139 
6-530344' 
653853-21 
6-596202 
6-6033541 
6-511489 
6-619106! 
6-626705' 
6-634237 
6641352, 
6-649400' 
6-656930; 
6-664444 
6-671940 
6-6794201 
6-635833: 
6-694329 
6-701759 
6-709173 
6-7165701 
6-723951 
6-7313161 
6-738664 
6-745997 
6-753313 
6-760614 
6-767899 
6-775169 
6-782423 
6-789661 
6-796834 
6-804092 
6-811235! 
6-8134621 
6 •3256-24 
6-83-2771 
6-839904 
6-847021 
6-854124 
6-861212 
6-868235 
6-875344 
6-882389 
6-889419 
6-896435 
6-903436 
6-910423 
6-917396 
6-924356 
6-931301 
6-933232 
6-945150 



336 
337 
333 
339 
340 
341 
342 
343 
344 
345 



112896 
113559 
114244 
114921 
115500 
116231 
116964 
117649 
118336 
119025 
119716 
120409 
121104 
121801 
122500 
123201 
123904 
124609 
125316 
126025 
126736 
127449 
128164 
128881 
129600 
130321 
131044 
131769 
132495 
133-225 
133956 
134689 
135424 
136161 
135900 
13764 
138.334 
139129 
139876 
1406-25 
141376 
142129 
142334 
143541 
144400 
145161 
145924 
146639 
147456! 
148225 
148996 
149769 
150544 
151321 
152100 
152831 
153564i 
154449, 
155-236 
„..., 1560251 

396 15'>816! 

397 157609! 

398 158404; 

399 159201! 
400! 160000 
40 Ij 150301! 
402! I6I6O4I 



347 

343 

349 

350 

351 

352 

353 

354 

.355 

356 

357 

353 

359 

360 

361 

352 

363 

354 

355 

365 

367 

363 

369 

370 

371 

372 

373 

374 

375 

376 

37 

378 

379 

380 

331 

332 

333 

334 

335 

386 

33 

333 

339 

390 

391 

392 

393 

394 

395 



Cube. 



Sq Root. CubeRoot- 



37933056 

33272753 

33614472 

38953219 

39304000 

39651821 

40001688 

40353607 

40707534 

410636-25 

41421736 

41781923 

42144192 

42508549 

42375900 

43243551 

43614208 

4398i977 

44361864 

44738875 

45118016 

45499293 

4588-2712 

46-268279| 

46G55000 

47045381 

47437928, 

47832147 

482235441 

43627125 

49027895' 

49430863; 

49835032! 

53243409i 

50653300; 

51054311 

51473348 

51895117 

5-2313624 

5-27343751 

531573761 

5358-2633 

54010152! 

54439939! 

54872000; 

553063411 

557429681 

56181887; 

56623104 

57066625; 

575l2456i 

57960603 

58411072 

53363359, 

59319000; 

59776471 

60235233! 

60693457| 

61162934 

6162^875 

6-2099135 

62570773 

630447921 

63521199: 

64000000; 

64481201 

64964808 



18-3303028 
183575598 
18-3847763 
18-41195-26 
18-4390889 

18 4661853 
18-4932420 
18-5202592 
18-5472370 
18-5741756 
18-6010752 
18-6279.360 
18-6547531 
18-6815417 
18-7082869 
18-7349940 
18-7616630 
18-7832942 
18-8148877 
18-8414437 
18-8679623 
18-8944436 
18-9208379 
18-947-2953 
18-9736660 
19-0000000 
19-026-2976 
19-0525589 
19-0787840 
19-1049732 
19-1311265 
19-1572441 
19-1833261 

19 2093727 
19-2353341 
19-2613503 
19-2373315 
19-3132079 
19-3390796 
19-3649167 
19-3907194 
19-4164878 
19-44-22221 
19-4679223 
19-4935887 
19-5192213 
19-5448203 
19-5703353 
19-5959179 
19-6214169 
19-64633-27 
196723156 
19-6977155 
19-7230829 
19-7434177 
19-7737199 
19-7939899 
19-8242276 
19-8494332 
19-8746069 
19-8997487 
19-9243588 
19-9499373 
19-9749844 
20-0000000 

20 0249344 
20 0499377 



6-952053 

6-958943 

6-965820 

6-97-2683 

6-979532 

6-936363 

6-993191 

7-000000 

7-006796 

7-013579 

7-020349 

7-027106 

7-033850 

7-040531 

7-047299 

7-054004 

7-06069' 

7-067377 

7-074044 

7-080699 

7-087341 

7-093971 

7-100533 

7-107194 

7-11.3737 

7-120367 

7-126935 

7-133492 

7-140037 

7-146569 

7153090 

7-159599 

7-166096 

7-172581 

7-179054 

7-185516 

7-191966 

7-193405 

7-204832 

7-211248 

7-217652 

7-224045 

7-23342' 

7-236797 

7-243156 

7-249504 

7-255341 

7-262167 

7-263482 

7-274785 

7-231079 

7-237362 

7-293633* 

7-299894 

7-306144 

7-312333 

7-318611 

7-324329 

7-331037 

7-337234 

7-3434-20 

7-349597 

7-355762 

7-361918 

7-363063 

7-374198 

7-330323 



APPENDIX. 



21 



Nc 1 Square. 


Cube. 


Sq. Root, 


Cubelioot.j! No. 


Square. 


Cube. 


Sq. Root. 


CubeRoot. 


403 


162409 


65450327 


20-0748599 


7-336437|i 470 


220900 


1033230^0 


21-6794334 


7-77493G! 


404 


163210 


65939264 


20-0997512 


7-392^42:: 471 


221841 


104487111 


21-70-25344 


7-73049.. 


405 


164025 


6643J125 


20-1246118 


7-398636!; 472 


222784 


105154048 


21-7255610 


7-785993 


406 


164836 


66923416 


20-1494417 


7-404721J 473 
7-4107y5 474 


223729 


105823317 


21-7435632 


7-791437 


407 


165 49 


67419143 


20-1742410 


224676 


106496124 


21-7715411 


7-79897-; 


408 ime-i 


67917312 


20-1990099 


7-416359 


1 475 


225625 


107171875 


21-7944947 


7-802454 


409 


167281 


63417929 


20-2237434 


7-4-22914 


! 476 


22r>576 


107850176 


21-8174242 


7-3079-25 


410 


168100 


68921000 


20-2484567 


7-423L'59 


! 4?7 


227529 


108531333 


21-840320; 


7-813389 


411 


168921 


69426131 


20-2731349 


7-434994 


1 478 


^2S484 


109215352 


21-8632111 


7-313346 


412 


16J744 


69934J 2i 


20-2977331 


7-441019 


1 479 


229441 


109902239 


21-8860836 


7-324294 


413 


17U5J9 


70444997 


20-32-24014 


7-447031 


1 4S0 


230400 


110592O0O 


21-908J023 


7-i;29735 


4U 


171396 


70957944 
71473375 


20-3469899 


7-453040 


! 4S1 


231361 


111281641 


21-9317122 


7-o3516j 


415 


172225 


20-3715483 


7-45 J03o 


1 4S2 


232324 


111930163 


21-95 4-;9 34 


7-810595 


41- 


17305!. 


71991296 


20-3950731 


7-465022 


! 4S3 


233239 


1 1-26785 i7 


21-9772610 


7-846013 


417 


173.S8J 


72511713 


20-4205779 


7-470999 


4M 


234258 


1133^9904 


22-OUOuOOO 


7-851424 


413 


174724 


73034632 


20-4450483 


7-476966 


! 4S5 


235225 


114034125 


22-0227155 


7-35882^ 


419 175531 


73580059 


20-46Mi95 


7-4329-24 


; 466 


236196 


114791256 


2-2'045 1-077 


7-33-2224 


420 


176400 


74083000 


20-4939015 


7-433372 


1 437 


237169 


115^01303 


22-0680765 


7-867813 


421 


177241 


74618461 


20-5182345 


7-494311 


i 438 


233144 


116214272 


220^07220 


7-872094 


422 


178084 


75151448 


20-5426336 


7-500741 


4^9 


239121 


116930160 


221133444 


7-87336-; 


423 


178929 


75636967 


20-5669638 


7-506661 


1 490 


210100 


117649000 


22 135943,; 


7-833 735 


424 


179776 


76225024 


20-5912603 


7-512571 


i 401 
492 


241031 


11837'j77i 


22-1535193 


7-8S9095 


425 


180625 


76765825 


20-6155281 


7-518173 


212064 


119095433 


22-l81u730 


7-894447 


426 


181 476 


77303776 


20-6397674 


7-524365 


493 


213049 


Il9823i57 


22-2033033 


7-899792 


4271 18232 J 


77854483 


20-6639783 


7-530243 


494 


244036 


1205537^4 


22-2-281 Kb 


7-905121 


423 


133184 


78402752 


20-6831609 


7-533122 


495 


245025 


1-21287375 


22-2485055 


7 9 1048'. 


429 


1840(1 


78953589 


20-7123152 


7-541987 


498 


246016 


122023i^36 


22-27105/5 


7-915783 


43U 


184930 


79507000 


20-7364414 


7-517342 


497 


247009 


r227634;3 


22-2034963 


7-921099 


431 


13^761 


80062991 


20-7605395 


7-553639 


! 498 


243004 


12-^5.59:72 


22-3150133 


7-ii2840-. 


432 136e*i4 


80621568 


20-7846097 


7-559526 


1 499 


249001 


124251499 


22-33:i3j79 


7-93 17k 


433; 187489 


81182737 


20-8086520 


7-535355I 500 


250000 


125000000 


22 -3600798 


7-^370.5 


434| 183356 


81746504 


20-8326667 


7-571174 


j 501 


251001 


1-25751501 


22-3330-293 


7-9422j3 


4.35! 18J^^5 


82312875 


20-8566536 


7-576985 


! 502 


252004 


126506008 


22 40535651 7-9475 /^i 


4361 190096 


82881856 


20-8306130 


7-532786 


503 


253009 


127263527 


22-42/6615 


7-95284b 


437] 190959 


83453453 


20-9045450 


7-583579 


504 


254016 


128024064 


22-4499443 7 9531141 


4331 191814 


84027672 


20-9284495 


7-504363 505 


255025 


12878762) 


22-472-2051 


7-9633/4 


439i 19272] 


84604519 


20-9523-263 


7-60013S| 506 


256036 


129554216 


22-4044433 


7-98362, 


440j 193600 


85134000 


20-9761770 


7-605905 i! 507 


257049 


130323343 


22-5186805 


7-973373 


441 194481 


8576 5121 


21-0000000 


7-611663 508 


253064 


13109O512 


22-533,i553 


7-9791U 


4421 195364 


86350388 


21-0237960 
21-0475652 


7-617412i! 509 


259081 


131872229 


22-5810^83 


7-984314 


443 196249 


86938307 


7-623152! 510 


260100 


132551000 


22-5331798 


7-939571 


444 197136 


87523384 


21-0713075 


7-6283841 511 


261121 


133432331 


22-8U53091 


7-994783 


445 198025 


83121125 


21-0950231 


7-634607il 512 


262144 


134217723 


22-6274170 


8 OOOOOt/ 


446 198916 


88716536 


21-1187121 


7640321! 513 


263169 


13500569/ 


22-8495033 


8-0052;!;: 


447 


199809 


89314623 


21-1423745 


7-646027JI 514 


264196 


135796744 


22-6 i 153^1 


8-01./403 


443 


200704 


89915392 


21-1660105 


7-651725 515 


265225 


136590575 


22-8038114 


8-0155'J.O 


449 201601 


90518349 


21-1396201 


7657414! 516 


266256 


137383096 


22 7158334 


8-020779 


450 202500 


91125000 


21-2132034 


7-663094 517 


267289 


13318^.413 


2^-73763401 8-0259571 


451 203401 


91733851 


21-2387606 


7-663766! 518 


263324 


138991832 


22-7500131 


8-0311-29 


452! 204304 


92345403 


21-2802916 


7-6744301 1 51ii 


269331 


13979835y 


22-7815715 


8-03829> 


453; 205209 


92959677 


21-2837967 


7-6800S6!! 520 


270400 


1406080JO 


22-8o35085 


8-041451 


454 206116 


93576664 


21-3072758 


7-685733!! 521 


271441 


1414-20761 


22-8^54244 


8-046603 


453; 207025 


94196375 


21-3307290 


7-69l372i! 522 


272434 


142236648 


22-84731b3 


8-051748 


456 


207936 


94818816 


21-3541555 


7-697002!' 523 


273529 


143055667 


22-0691933 


8 056836 


457 


208849 


95443993 


21-3775533 


7-702625! 524 


274576 


143877821 


22-8910463 


8-062018 


458; 209764 


96071912 


21-4009346 


7-708239,; 525 


275625 


144703U5 


22-;il28785 


8-067143 


459i 210581 


96702579 


21-4242353 


7-713345 ' 526 


276676 


145531576 


22-9346 S99 


8-072262 


4601 211600 


97336U00 


21-4476106 
21-4709106 


7-719443; 5-27 


277729 


148363183 


22-9584306 


8-077374 


461! 212521 


97972181 


7-725032' 523 


278784 


147197952 


22-9732508 


8 082430 


462^ 213444 


98511123 


21-4941353 


7-7306 14 i' 529 


279341 


148035389 


23 0000000 


8-087579 


463; 214369 


9925 i347 


21-5174318 


7-736183'; 530 


280900 


148877000 


23-0217^89 


8-09267^ 


464| 215296 


99897344 


21-5406592 


7-741753; 531 


281961 


149721-2^1 


23-0434372 


8-097759 


465 216225 


100544625 


21-5633537 


7-74731 1^ 532 


233024 


150568763 


23-0651-252 


8- 10-233 J 


466 217156 


101194896 


21-5370331 


7-7528611 533 


234089 


151419437 


23-0887928 


8-107913 


467! 218; y9 


101847563 


21-6101828 


7-753402' 531 


•285156 


152273304 


231084400 


8-112931. 


1 468! 219024 


10i503232 


21-6333077 


7-763j361 535 


236225 


153130375 


23-1300870 


8-1 1^04 1 


1 469 219961 


, 10 5161709 


21-6561078 


7-769462': 536 


287296 


153990656 


23-1516738 


8-12309., 



22 



APPENDIX. 



No. Square. 



Cube. 



. Root. CuheRoot- ! No. I Square. Cube. 



5i4| 
545 
5i6! 

547j 
54S 
5191 



537 233369 
5331 2394441 
533| 29J5il| 
510! 291600 
.541, 292631' 
5121 293764; 
513 294349: 
2959361 
2970251 
2981161 
2992391 
3003041 
301401} 
550' 3025:JO: 
551' 3)3601j 
552) 3J47()4| 
553' 3J5S09 
554i 306916 
555^ 303025 
556 309136 
\ 5571 310249 
553 311364 
559 312431 
560, 313600 
561 314721 
562' 315344 
563' 316969 
564 313096 
5'5: 319225 
566: 323356 
5'37| 32U39I 
533; 322624! 
569 3^3761 
570' 324900 



o 
572 



326011 
327134 



573 329329! 

574 329476; 

575 330625' 

576 3317761 

577 332929' 
573: 334034' 
579' 33524]! 
530^ 3354 JO ' 
53ii 337531 
532 33i724' 
533' 3393391 
^34 341055; 
5^5' 342:225 
536 313396 
5"^7^ 3145391 
535' 345744i 
539; 316921' 
59U 343100. 
591 349281 
'^.>2' 350464 
593' 351649 
594! 352S36 
5951 354025 
596| 355216 
5971 356409 
5981 357604 
5991 353S01 



360000 
361201 
362404 
3636! 59 



154854153 

155720872 
156590819 
1574640C0 
153340421 
15.^220083 
160103007, 
160939134 
161S73625 
162771336 
163667323 
161566592 
165169149| 
163375000 
1672S4151 
163196608; 
169112377! 
170031454! 
170953375; 
171879618' 
172308693; 
I7374III2I 
174676879! 
1756 16000 1 
176553481 
1775.U323: 
173453547 1 
1794;)6144; 
130362125: 
181321496; 
182234263| 
133250432 
18422v>009 
185193000 
136I6j4nl 
137 14 J2 181 
18S13.;517J 
189119224 
190109375 
191l02y7t;>l 
19210,10331 
193100552 
194104539 
195112000 
lL-6122941 
197! 37363 
1931552S7 
199176704 
200201625 
201230056 
202202003 
v:03297472 
204S33l6.t 
20537s000 
206425071 
207474{;33 
2085i7357 
2095 34 ■)81 
21064487; 
21170373(. 
212773173 
213347192 
214921799 
216000000 
217081301 
218167208 
219256227 



2317326051 
23-1943270; 
23-21G3735! 
23-43790011 
23-25910671 
23-2S03935 
23.3023604 j 
23-3233076 
23-3452351! 
23-3663429! 
23-33803111 
23-4093998] 
23-4307491^ 
^3-4520788 1 
23-4733392' 
23-4946802 
23-5159520| 
235372046' 
235534330| 
23-5796522! 
23-6003474! 
23 6-220236' 
23-6431808 
23-6643191! 
23-6354336! 
23-7065392! 
23-7276210 
23-7486342: 
23-7697-283! 
23-7907515' 
23-3117618! 
23-S3275J6 
23-S537209 
23-8746728; 

23 8.<560o3 
23-9135215 
23-J374184 
23-9532j71| 
23-97915761 
24-(»u000J0| 
2102082431 
21-04 16306; 
24-0624188 
24-0d31891; 
24-1039416 
24-1216762 
24- 145392 J; 
24-1660919! 
24-1867732! 
242J74339; 
24-2230829' 
212187113 
24-2393222 
24-2399 156; 
24-3104916 
213310501 
24-3515913 
24-3721152 
24-3.;232l3 
21-4131112 
24-4335334! 

24 45:0335 
•^1.4744765 
24 4943974! 
24-5i53Jl3 
21-5356383 
24-55605 33i 



8-123145! 

8-133187 

8-133223 

8-143253 ' 

8-148276 1 

8-153294 

8-158305 ■ 

8-163310 

8-168309 

8-173302 

8-178239 

8-183269 

8-188244 

8-193213 

8-198175 

8-203132 

8-203082: 

8-213027'! 

8-217966' 

8-222893 

8-227825: 

8-^32746 

8-237661 

8-24-2571 i 

8-217474! 

8-252371 

8-257263! 

8-^62149: 

8-267029 

8-271904 

8-276773' 

8-281635 

8-236493' 

8-291344; 

8-2.;019u! 

8-3J103. 

8-3J5365 

8-310694 

8-315517 

8-32 j3 '5 ; 

8-325117' 

8-329954i: 

8-33 17551 

8-3395511 

8-34 13411 

8-349126 

8-353i;05; 

8-35S678' 

8 3334 17: 

8 363-209' 

8-372967! 

8-3777 1j| 

8-332165 

8-337206; 

8-391942! 

8-396673! 

8-401 398' 

8-406 1 1-V 

8-410333. 

8-415542 

8-42024i)i 

8-424945 

8-429633! 

8-431327; 

8-4390 10| 

8-44368ii 

84183601 



607 
608 
609 
610 



613 
611 
615 
616 
617 



601 3.4816 
605i 3.6025 
606! 3=-.723e 
36844- 
369664 
370881 
372100 
Oil' 373321 
612! 371554 
3757n9 
376996 
378225 
379156 
33068'J 
618' 331924. 
619' 383161! 
6201 33140O| 
621 3356411 
622! 38683;! 
623 38^1291 
624! 339376; 
6-25. 390625 
626 39i87t;i 
627; 393129! 
62i'. 394334J 
629! 3^5641 ! 
63o; 396900 1 
631: 3J316II 
632- 399424I 
633' 40068.' 
4J1956 
4^3225 1 
636' 404!96| 
63/1 405769 
633: 407u44i 
639 4U832li 
6I0! 40j6 KJ 
641 4 10881 1 
642: 4121641 
643i 413419! 
61.! 414733i 
645; 4l60i:5i 



Sq. Rcct. CiiteRoot 



6^ 
635 



616 



417316 
^^47j 4133091 
618, 419J04I 
649' 4212v«I; 
659 4225 J^H 
651' 4233^11 

652 425104! 

653 423409; 

654 4-27716! 

655 429025' 

656 43 336' 

657 4316*9! 
653 432934' 
659; 43423 1 1 
66' I, 43560o! 
66 1! 436921' 

662 438244 

663 439539' 
661 440393! 

665 442225: 

666 443^)56: 

667 444389: 
663 416,21; 

669 447531 

670 44890;j' 



220348364 
2214451:^5 
222545016 
223648543 
224755712 
225865529 
^26981000 
228099131 
229220928 
23oai6397 
231475544 
232608375 
2337448961 
2348851 13| 
233029032 1 
2371760591 
233328-00! 
239433061 
240641818! 
211304367; 
242970624! 
21414^625] 
245314376! 
246491833 
247673152 
243853189 
250017000 
251239591 
252135968 
253636137 
254810101 
256047375 
257259456 
258174853 
259694072 
260917119 
262144000 
2()3374721 
234609283i 
265847707 
267089984 
238336125 
269586136 
270810^.123 
272097792 
273359449J 
2746-25 JOO! 
2758944511 
277ir)7808; 
273445077; 
279723264' 
281011375 
282300416^ 
283593393' 
2343903 12 
235191179 
2374960<J0 
2338047S1 
290117528 
291434247 
292751941 
294079625 
295408296 
296740963 
298077632 
299413309 
390763000 



24-5764115 8-453028 
24-5967478, 8457691 
24-6170673, 8462348 
21 63737001 8-46700C 
24-6576560i 8-471647 
24-6779254! 8-4762.99 
24 69817811 8-480926 
24-71841421 8-485558 
24-7386338| 8 490185 
24-7538338| 8-494806 
24-7790234 8-499423 



24-7991935 
24-81934-73 
24-8394347 
24-8596058 
21-8797106 
21-8997992 
21-9193716 
24-9399278 
24-959. 679 
24-979992(1 



8-594035 
8-508642 
8-513243 
8-517840 
8-522432 
8-52701S 
8-531601 
8-533178 
8-540750 
8-545317 



25-00000001 8-51988! 



25-019992. 
25-039963; 
25-059928:^ 



8-551437 
6-553990 
8-563539 



I .^ 



25-0798724 8-56308 
25099300s 8-572619 
25-11971341 8577152 
25 139S10-2I 8-;38l631 
•^5 1591913' 8-586205 
25-17935661 8 5'.}0724 
25-19920631 8-59.5233 
25-21904041 8-5-99748 
25 2333539! 8-604252 
25-2.5356191 8-6J8753 
25 2731493! 8-613248 
25.2932213; 8 -6 17739 
25-3179778' 8-62-2225 
•^5-3377189| 8-626706 
25-35744471 8-631183 
25-3771551 8-635655 
•^5 3968502 8.640123 
25-41653^1 8-644585 
25-43 31947| 8-649044 
25-4.5531411 8-653197 
25 4754734, 8-657946 
25-49509761 8-632391 
25 51470161 8-66693] 
25 5342907J 8-671266 
25-5533647 8 67569 
25-5734237; 8-680124 
25-5929578; 8684546 
25-61-24969; 8-688963 
25-6320112 8-693376 
25-65151071 8-697784 
25-6709953 8702188 
25 6901652 8-706.588 
25-7099203 8710983 
25-7293607: 8-715373 
25-7487834 8-71976( 
25-7631975 8721141 
25-7875939 8-728518 
25-8069758 8-732892 
25-826^131! 8-737260 
25-815606)! 8 741625 
25-8550343! 8-745985 
25-8943582: 8-75 saio 



APPENDIX. 



23 



No, 


Square. 


Cube. ! Sq. Root. IcubeRool. 


No. 


Square. Cube. 1 


Sq. Roct. 


CubeRoot. 


67 J 


450241 


302111711 


25-9036677 8-754691 


738 


544644 


401947272 


27-1661554! 


9-036888 


672 


451584 


303464448 


25-9229628 S-759033 


739 


546121 


403533419 27-1845544 


9040965 


673 


452929 


304821217 


25 •942-2435 8-783331 


740 


547600 


405224000 27-2029410 


9-045042 


674 


454276 


3081320^4 


25-9615100 


8-787719! 


741 


549081 


406369021 27-2213152 


9-049 ll^i 


675 


455825 


307546375 


25-9807821 


8-7720531 742! 550564| 


403518438 27-2398789 


9-0531o3 


676 


45697*. 


308915778 


28-0000000 


8'778333j 


743 552049 


41017-2407 27-2580283 


9-057243 


6771 45SJ2y 


310283733 


280192237 


8-780708 


744 553536 


411830754 27-27638341 


908131U 


678 45%S4 


311665752 


26-0334331 


8-785030 


745 555025 


413493825! 27-294888^ 


9-085383 


87y 


461041 


313048839 


26.0576284 


8-789347 


746i 556518 


41516U;i38l 27-3130006 


9-069422 


6S0 


46240U 


314432000 


2O-0763096 


8-7938591 


747 558009 


4168327231 


27-3313007 


9-073473 


681 


463761 


315821241 


26-0959787 


8-797968i 


743 559504 


418508992 


27-3495387 


90775^0 


6S2 


46J12^ 


31721-1588 


26-1151297 


8-80-22721 


749 


581001 


420189749 


27-3878644 


y -031563 


683 


468489 


313811987 


26-1342687 


8-8085721 


750 


562500 


421375000 


27-3o6l279 


9-085003 


6M 


467856 


320013504 


26-1533937 


8-810888! 


751 


564001 


423564751 


27-4043792 


9-03y633 


685 


469225 


321419125 


26-1725047 


8-8151801 


752 


585504 


425259008 


27-4228184 


9-Or;3672 


886 1 4705% 


322828858 


26-1916017 


8-819447 


753 


5u7009 


428957777 


27-4403455 


9-097701 


687 i 47 i 'Job 


3^4242703 


26 2106848 


8-82373li 


754 


583518 


423861064 


27-4590604 


y-10l7i:b 


63S 


473344 


3ii508o672 


28-2297541 


8-823010 


755 


570025 


430368875 


27-4772633 


9-105748 


639 


474721 


3^7082/89 


23-2483095 


8-832235 


75b 


571538 


432081210 


27-4954542 


9-109787 


690 


476100 


3235 J9i '00 


26-2673511 


8-838558 


757 


573049 


4337980y3 


27 513833^ 


y-113732 


691 


477431 


32yy39.-i71 


28-2888789 


8-840823 


758 


574584 


435519512 


27-5317998 


9-1177y3 


69z 


478^6. 


33137333b 


i6-305392'j 


8345085 


7o^ 


57ou81 


437245479 


27-549y548 


9-121301 


6931 48U2ib 


33^812557 


28-3248932 


3-84^-344 ' 760 


577800 


433976000 


27-5080975 


9-125305 


t>94 


481830 


3o4255334 


28 -34387^7 


8-853598 


761 


57yl2l 


440711081 


27-5302234 


9-12y8b6 


695 


433J25 


335702;i75 


28-3628527 


8-857849 


782 


530644 


442450728 


27-6043475 


913330b 


696 


484410 


337153538 


26-3318119 


8-88-2095 


783 


582169 


444194947 


27 6224546 


y-i377.;7 


697 


435309 


33cj603873 


28-4007576 8.86o337 


784 


533096 


445943744 


27-64U549;; 


y-141787 


698 


487204 


340063392 


26-4196896 


3-870578 


765 


535225 


447897125 


27-6580334 


9-14577^ 


699 


483601 


34153-099 


26-4386081 


8-874810 


768 


588750 


44^455096 


27b7o7o50 


y-14y75o 


7U0 


490000 


343000000 


26-4575131 


8-879040 


787 


538289 


451217663 


27-8947640 


9-153737 


701 


491401 


344472101 


26-4764040 


8-883286 


783 


589824 


452984832 


27-7128129 


yl577i<4 


702 


492C504 


345943408 


26-4952826 


8-88748S 


769 


5Jl38i 


454756809 


27-730849^ 


9-181037 


7U3 


494209 


347428927 


28-5141472 


8-891706 


1 770 


592900 


4585^3000 


27-7488739 


yl8585o 


704 


495616 


34891368 i 


26-5329y83 


8-895920 


1 771 


594441 


458314011 


27-7668888 


9169U2, 


705 


497025 


35040i625 


28-5518381 


8-900130 


77<i 


5j5y84 


480099848 


27-7843880 


9-17353^ 


706 


493436 


351895316 


26-5706605 


8-904337 


773 


597529 


46 188991 7 


27-8023775 


y-l775i'i 


707 


499849 


353393243 


26-5394716 


8-908539 


774 


599078 


463884824 


27-8208555 


9-18150L 


708 


501264 


354894912 


26 •6082694 


8-912737 


775 


600825 


485484375 


27-8333218 


y-i8545o 


709 


502681 


356400829 


26-6270539 


8-918931 


776 


602176 


467288578 


27-8587760 


9-18^*0^ 


710 


504100 


357911000 


26-8458252 


8-921121 


777 


603729 


489097433 


27-8747ly7 


y-ly3347 


711 


505521 


359425431 


26 6645833 


8-925308 


778 


605284 


470910952 


27-8926514 


yi9729o 


712 


506944 


38094412b 


26-6833231 


8-929490 


77y 


608841 


47V729139 


27-9105715 


9-20l22y 


713 


508389 


362487097 


28^70205J8 


8-933689 


78o 


6034wO 


474552000 


27-9284801 


9-2051O1 


714 


50»796| 363994344 


28-7207784 


8-937843 


781 


60jy61 


47t37y541 


:^7-y403772 


y-2ov.0-./t 


7l5i 5112251 365525375 


k-^ -7394839 


8-942014 


782 


6115^4 


47o2ll768 


27-9842829 


9-2l30iio 


716 


512656 367081896 


26-7581703 


8-946181 


1 783 


613089 


480048687 


27-982137^ 


y-2loy5^^ 


717 


514089! 368601813 


26-7788557 


8-950344 


784 


614056 


481890304 


28-oooouou 


9-220873 


718 


515524 


370148232 


26-7955220 


8-954503 


785 


616225 


433736825 


28 0173515 


y-2247yi 


719 


516%1 


371894959 


28-81417541 8'958858 


786 


8177y8 


435537658 


28-0356yl5 


9-223707 


720| 518400 


373248000 


28^8328157 


8-982809 


1 787 


61936y 


437443403 


28-0535203 


9-2^2619 


721 519341 


37480538] 


28-3514432 


1 8-988957 


j 788 


620944 


43y303372 


28-0713377 


9-2bb52b 


722 521284' 3/b3u70-xH 


26-8700577 


1 8-971101 


1 789 


622521 


491169089 


28-039 l43i 


9-240435 


723 5227291 377933o67 


28^8888593 


8-975241 


! 790 


8i.4100 


4y3o3yoo0 


23-108y336 


9-Vi44c3o 


724| 52il76 3795L3424 


269072481 


! 8-979377 


! 791 


625681 


4y4913671 


28-1247222 


9-243234 


725 


525825! 331078125 


26-9258240 8-983509 


i 792 


627284 


498793083 


28-l424y4b 


y -252 130 


726 


527078 382657176 


26-9443872 8-987637 


793 


628849 


4^3677257 


28-1602557 


•j-zbbU'ZZ 


727 


528529 334240583 
52J984' 38582835^ 


26-9629375 8-991762 


1 794 


630436 


500506184 


28-1780050 


9-25yyii 


728 


28-9814751 8-995833 


1 795 


632025 


502459375 


23-1957444 


9-26.wyV 


729 


531411! 337420489 


27-0000000 9-OUOOOO 


1 796 


63361b 


504353336 


23-2134720 


9-2b708o 


73U 


532y00i 389017000 


27-0185122 9-0U4113 


i 79< 


63520y 


50826 15 <o 


23-231 io34 


'JzliOOi, 


731 


5343611 390617891 


27-03701171 9-008223 


1 798 


636804 


508189592 


28-2435y3o 


y -270430 


73^ 


1 535824! 392223168 


27-055-i985' 9-012329 


1 799 


638401 


5100823yy 


28-2805331 


9-;47y3j3 


733 


5272H9I 393832837 


27-07 3'j727 9-016431 


800 


640000 


512000000 


28-2842712 


y-233l73 


734 


53o7;'6 3j54469i.>4 


27-0924344 9-020529 


! 801 


641601 


513922401 


23-3019434 


9-237041 


735 


540225' 397065375 


27.1108834: 9024624 


1 802 


64320-4 


51534y808 


23-319(,045 


y-;i90.u7 


736 


1 5-llt.9(.! 393688250 


27-1293199! 9-0-^8715 


1 803 


844809 


517781827 


28-3372548 


9-2y4(0. 


737 


j 5431t>9! 400315553 


27-147743y 9-032802 


i 804 


646416 


519718464 


28-3543933 


9-2y3biT 



u 



APPENDIX. 



No. 


Square. 


Cube. 1 Sq. Root. CubeRoot.' 


No. 1 


Square. 


Cute. . 


Sq. Root 


CubeRoot. 


805 


648025 


521660125 28 3725219 


9-302477i 


"872 


760384 


663J54S48I 


29-5296461 


9-553712 


806 


649636 


52360C616 28-3901391 


9-306328! 


873 


762129 


655338617 


29-5465734 


9-557363 


807 


651249 


5255579431 28-4077454 


9-310175 


874 


763376 


667627624 


29-5634910 


9-561011 


808 


6528G4 


527514112' 28 4253408 


9-3140191 


875 


765625 


669921875 


29-5303989 


9-564656 


809 


654481 


52947 5 129 1 23-4429253 


9-317860 


876 


767376 


^72221376 


29-597-2972 


9-568298 


810 


656100 


5314410001 ii8-46'J4989 


9-321697 il 877| 


769129 


674526133 


29-6141853 


9-571938 


811 


657721 


533411731 23-4780617 


9-3-25532' 


878 


770884 


676336152 


29-6310648 


9-575574 


812 


659344 


5353873281 23-4956137 


9-329363 


879 


772641 


679151439 


29-6479342 


9-579208 


813 


660969 


537367797 23-5131549 


9-333192! 


880 


774400 


.681472000 


29-6647939 


9-53-2840 


Gl4 


662596 


539353144 


23-53068521 


9-3o7017| 


531 


776161 


6b3797841 


29-6816442 


9-:86468 


815 


664225 


541343375 


23-5482048 


9-340839: 


832 


777924 


6861289681 


29.6984843 


9-590094 


816 


665S56 


543338496 


23-5657137 


y-3;4657| 


883 779639 


688465387 29-7] 531591 9-5937171 


817 


667489 


54533^513 


28-5332119 


9-3-:8473; 


8i4| 781456 


69U807104 


29-7321375 9597337 


818 


669124 


547343432 


28-6006993 


9-352236' 


8-j5 


733225 


6^31541-25 


29-7489496 9-600955] 


819 


670761 


549353259 


28-6181760 


9-356095; 


SS6 


78499^ = 


695506456 


29-7657521 


9-604570 


820 


67240(i 


551368000 


28-6356421 


9 359902! 


887 


78S769 


697864103 


29-7825452 


9-608182 


821 


674041 


553387661 


28-6530976 


9-363705! 


888 


733544 


700227072 


29-7993239 


9-61179! 
9-6 1539'- 


822 


675684 


5554122481 28-6705424 


9 367505; 


839 


790321 


702595369 


29-816103J 


823 


677329 


^57441767 23-6379766 


9-371302 


890 


7^2100 


704969000 


29-8328678 


9-619002 


824 


678976 


559476224! 23-7054002 


9-375096| 


891 


793381 


707347971 


29-8496231 


9-622603 


825 


680625 


561515625 23-7228132 


9-373387 


892 


795664 


709732288 


29-8663690 


9-6-26-202 


826 


6S2276 


563559976 


28-7402157 


9-3326751 


893 


797449 


712121957 


29-8831056 


9-629797 


«27 


633J29 


565609283 


28-7576077 


9-336460! 


894 


799236 


714516934 


29-8998328 


9-633391 


823 


685584 


567663552 


28-7749391 


9-390242' 


895 


801025 


716917375 


29-9165506 


9-636981 


829 


687241 


569722789 


28-7923601 


9-394021! 


896 


802816 


719323135 


29-y332591 


9-64056S 


830 


6839,M., 


571787000 


28-8097206 


9-3.7790;i 897 


804609 


721734-273 


29-9499533 


9-644154 


831 


690561 


573356191 


28-8270706 


9-401569! 


893 


806404 


724l507;-;2 


29-9666481 


9-647737 


832 


69i224 


57593J368 


28-8444102 


9-405339^ 


899 


803201 


726572699 


29-9833237 


9-651317 


833 


693889 


578009537 


28-8617394 


9-409105! 


901) 


810000 


729000000 


30-0000000 


9-654894 


83-4 


695555 


5S0093704 


28-8790582 


9-412869 1 901 


811801 


73143-2701 


300166620 


9-658468 


8b5 


697225 


532182875 


2M-89636G6 


9-416630: 


902 


81S604 


733870808 


30-0333148 


9-662040 


«o6^ 693896 


534277056; 28-9136646 


9 -4-20337 1 


903 


815109 


7363143-27 


30-0499584 


9-665610 


837 


700569 


586376253 28-9309523 


9-424142! 


904 


817216 


738763-264 


30-0665928 


9-659176 


838 


702244 


5^8480472 28-9482297 


9-4278 14: 


905 


819025 


741217625 


30-0832179 


9-672740 


839 


703921 


590539719 23-9654967 


9-431642 


906 


820836 


74.^677416 


30-0998339 


9-676332 


840 


705600 


5927040001 28-9827535 


9-435338 


907 


822649 


746142643 


30- 1164407 


9-679860 


8il 


707281 


59482332i| 290000000 


9-439131 


908 


824464 


748613312 


30-1330383 


9-683417 


8i2 


708:64 


596947688! 29-0172363 


9-442370 
9-446607 


909 


826281 


751089429 


30-1496269 


9-636970 


• 8i3 


710649 


599077107 29-0344623 


910 


828100 


753571000 


30-1662063 


9-690521 


844 


712336 


60121 15S4I 29-0516781 


9-450341 


911 


8-29921 


756053031 


30- 18-27765 


9-694069 


845 


714025 


603351125 29-0688837 


9-454072 


912 


831744 


758550528 


30-1993377 


9-697615 


816 


; 715716 


605495736 29-0860791 


9-457800 


913 


833569 


761048197 


30-2158399 


9-701158 


817 


717409 


607645123 29-1032644 


9-461525 


91^ 


835396 


763551944 


30-2324329 


9-704699 


848 


719104 


609300192 29-1204396 


9-465247 


915 


837225 


76fi(;60375 


30-248966'J 


9-708237 


849 


! 720S01 


6119600491 29-1376046 


9-463966 


•916 


839056 


768575296 


30-2654919 


9-711772 


850 


i 722500 


614125000 29-1547595 


9-472682 


917 


84088t 


771095213 


30-2820079 


9-715305 


851 


i 724201 


616295051 29-1719043 


9-476396 


913 


84272^ 


7736-20632 


3-)-2985148 


9-718835 


852 


1 725904 


618470203: 29-1890390 


9-480106 


919 


844561 


776151559 


30-3150 1-28 


9-722363 


853 


! 727609 


620650477 


29-2061637 


9-483814 


92. 


846401 


773688000 


3O-3315018 


9-725888 


854 


72931f 


622835864 


29-2232784 


9-4d7518 


921 


848241 


781229961 


30-3479818 


9-729411 


855 


7310^5 


625026375 


29-240383J 


9-491220 


922 


850081 


783777448 


30-3644529 


9 732931 


856 


7327 3e 


627222016 


29-2574777 


9-494919 


923 


85192t 


1 786330467 


30-3809151 


9-736448 


857 


73444t 


629422793 


29-2745623 


S -4986 15 


924 


8537761 788889024 


30-3J73683 


9-739963 


858 


736164 


631623712 


1 29-2916370 


9-5023J8 


925 


8556-25 791453125 


30-4138127 


9-743476 


859 


i 737881 


633339779 


! 29-3087018 


9-505998 


926 


857476! 794022776 


3J-43.)2481 


9-746986 


860 


73960C 


636056000! 29-3257566 


1 9-509G85 


9^; 


859329' 796597983 


30-4465747 


9-750493 


861 


741321 


6382773811 29-3423013 


1 9-513370 


928 


861184' 79917875-.i 


30-4630924! 9-753998 


862 


743044 


64050392S! ^9-3598365 


! 9-517051 


929 


ft03041 801765089 


30-47950 131 9-757500 


863 


i 744769' 6427356471 29-3768616 


! 9-520730 


93u 


8649001 804357o0( 


30-49590141 9-761000 


834, 7464961 614972544! 29-393376 J 


! 9-524406 


931 


866761! 80695^491 


1 30-5 12-29261 9-764497 


865' 748^25 


647214625; 29-410^823 9-528079 


932 


863n2.l. 809557568 


30-5-286750! 9-767992 


ii6:>. 74995( 


) 649461896! 29-4278779; 9-531750 


933 


8704--.>j 812166237 


1 30-5150487J 9-771484 


867; 751681 


651714363' 29-4448637 9-535417 


9ci4 


87235G; 8147'80504 


30-5614136! 9-774974 


86fc 


; '/ 5342-5 


[ 6539720321 29-461b397 9-539J82 


935 


! 874225' 817400375 


30-5777697 


9-7734621 


86'. 


. 75516] 


656234909 29-47.-i8059 9-542744 


936 


876096! 820025y56 


30-5'.;41i71 


9781947! 


a7c 


75690( 


J 6585030001 29-4'j57624 9546403 


937 


877969| 82-2656953 


30-610455? 


i 9'7854-29; 


S7] 


1 75864 


660776311; 2J-51270ei: 9-550059 


93^ 


8798441 825293672 


30-62678571 y-78-908J 



APPENDIX. 



26 



No. 


Square. | Cube. 


Sq. Root. 


CubeRoot.' 


No. 


Square. 


Cube. 


Sq. Root. 


CubeRoot. 


939 


8817211 827936019 


30-6431069 


9-792386 


970 


940900 


912673000 


31-1448230 


9-898983 


940 


883600 


830584000 


30-6594194 


9-795861 


971 


942S41 


915498611 


31-1608729 


9-902383 


941 


885481 


833237621 


30-6757233 


9-799334 


972 


944784 


918330048 


31-1769145 


9-905782 


942 


887364 


835896888 


30-6920185 


9-802804 


973 


946729 


921167317 


31-1929479 


9-909173 


943 


889249 


838561807 


30-7083051 


9-806271 


974 


9486761 924010424 


31-2039731 


9-912571 


944 


891136 


841232334 


30-7245S30 


9-809736 


975 


950625 


926859375 


31-2249900 


9-915962 


945 


893025 


843908625 


30-7408523 


9-813199 


976 


952576 


929714176 


31-2409987 


9-919351 


946 


894916 


846590536 


30-7571130 


9-816659 


977 


954529 


93-2574833 


31-2569992 


9-922733 


947 


896809 


849278123 


30-7733651 


9-820117 


978 


956484 


935441352 


31-2729915 


9-926122 


948 


898704 


851971392 


30-7896086 


9-823572 


979 


958141 938313739 


31-2889757 


9-929504 


949 


900601 


854670349 


30-8058436 


9-827025 


980 


960400 941192000 


31-3049517 


9-932884 


950 


902500 


857375000 


30-8220700 


9-830476 


981 


962361 944076141 


31-3-209195 


9-936261 


951 


904401 


860085351 


39 8382879 


9-833924 


982 


9643241 946966168 


31-3368792 


9-939635 


952 


906304 


862801408 


30-8544972 


9-837369 


933 


966289 949862087 


31-3528308 


9-943009 


953 


908209 


865523177 


30-8706981 


9-840813 


934 


968256 952763904 
970225 955671625 


31-3687743 


9-946330 


954 


910116 


868250664 


30-8868904 


9-844254 


985 


31-3847097 


9-949748 


955 


912025 


870983875 


309030743 


9-847692 


986 


9721961 958535256 


31-4006369 


9-953114 


956 


913936 


873722816 


30-9192497 


9-851128 


987 


974169| 961504803 


31-4165561 


9-956477 


957 


915849 


876467493 


30-9354166 


9-854562 


288 


9761441 9644302^2 


31-4324673 


9-959839 


958 


917764 


879217912 


30-9515751 


9-857993 


989 


9781211 967361669 


31-4483704 


9-963198 


959 


919681 


881974079 


30-9677251 


9-861422 


990 


9801001 970299000 


31-4642654 


9-966555 


960 


921 SCO 


884736000 


30-9838668 


9-864848 


991 


982081 


973242271 


31-4801525 


9-969909 


961 


923521 


887503681 


31-0000000 


9-868272 


992 


984064 


976191488 


31-4960315 


9-973262 


962 


925444 


890277128 


31-0161248 


9-871694 


993 


986049 


979146657 


31-5119025 


9-976612 


963 


927369 


393056347 


31-0322413 


9-875113 


994 


988036 


982107784 


31-5277655 


9-979960 


964 


929296 


81)5841344 


31-0483494 


9-878530 


995 


990025 


985074875 


31-5436206 


9-983305 


965 


931225 


898632125 


31-0644491 


9-881945 


996 


992016 


988047936 


31-5594677 


9-986649 


966 


933156 


901428696 


31-0805405 


9-885357; 


997 


994009 


991026973 


31-5753068 


9-989990 


967 


935089 


904231063 


31-0966238 


9-88S767; 


998 


996004 


994011992 


31-5911330 


9-993329 


\968 


937024 


907039232 


31-1126984 


9-892175 


999 


998001 


997002999 


31-6069613 


9-996666, 


'969 


938961 


•909853209 


31-1287648 


9-895580 


1000 


1000000 i 1000000000 


31-6227766 


lO-OOOOOOJ 



The following rules are for finding the squares, cubes and roots, of 
numbers exceeding 1,000. 

To find the square of any number divisible without a remainder. 
Rule. — Divide the given number by such a number, from the forego- 
ing table, as will divide it without a remainder ; then the square of the 
quotient, multiplied by the square of the number found in the table, 
will give the answer. 

Example.- -What is the square of 2,000 ? 2,000, divided by 1,000, 
a number lound in the table, gives a quotient of 2, the square of which 
is 4, and the square of 1,000 is 1,000,000, therefore : 
4 X 1,000,000 = 4,000,000 : the Ans. 

Another example. — What is the square of 1,230 ? 1,230, being di 
vided by 123, the quotient will be 10, the square of which is 100, and 
the square of 123 is 15,129, therefore : 

100 X 15,129-= 1,512,900: the Ans. 

To find the squafe of any number not divisible without a remainder. 
Rule. — Add together the squares of such two adjoining numbers, from 
the table, as shall together equal the given number, and multiply the 
sum by 2 ; then this product, less 1, will be the answer. 

Example. — What is the square of 1,487 I The adjoining numbers, 
743 and 744, added together, equal the given number, 1,487, and the 
square of 743 = 552,049, the square of 744 == 553,536, and these 
added, = 1,105,585, therefore : 

1,105,585 X 2 = 2,211,170 — I -= 2,211,169: the Ans. 

To find the cube of any number aivisihle without a remainder. 
Rule. — Divide the given number by sucn a number, from the forego- 



20 APPENDIX. 

ing table, as will divide it without a reinainder ; then, the cube of tne 
quotient, multiplied by the cube of the number found in the table, will 
give the answer. 

Example.— \Sf\\d.\ is the cube of 2,700 ? 2,700j being divided by 900, 
the quotient is 3, the cube of which is 27, and the cube of 900 is 
729,000,000, therefore : 

27 X 729,000,000 -= 19,683,000,000: the Ans. 

To find the square or cuhe root of numbers higher than is found in the 
table. Hide. — Select, in the column of squares or cubes, as the case 
may require, that number which is nearest the given number ; then 
the answer, when decimals are not of importance, will be found di- 
rectly opposite in the column of numbers. 

Example. — Wliat is the square-root of 87,620 ? In the column of 
squares, 87,616 is nearest to the given number ; therefore, 296, im- 
mediately opposite in the column of numbers, is the answer, aearly. 

Another example. — What is the cube-root of 110,591 ? In the co- 
lumn of cubes, 110,592 is found to be nearest to the given number ; 
therefore, 48, the number opposite, is the answer, netirly. 

To find the cube-root more accurately. Rule. — Select, from the co- 
lumn of cubes, that number which is nearest the given number, and 
add twice the number so selected to the given number ; also, add twice 
the sriven number to the number selected from the table. Then, as 
the former product is to the latter, so is the root of the numbeir selected 
to the root of the number given. \ • 

Example. — What is the cube-root of 9,200 ? The nearest number 
in the column of cubes is 9,261, the root of which is 21, therefore : 
9261 9200 

2 2 



18522 18400 

9200 . 9261 



As 27,722 is to 27,661, so is 21 to 20-953 -i- liie Ans. 

Thus, 27661 X 21 = 580881, and this divided I7 27722 = 20'953 + 

To find the square or cube root of a whole number with decimals. 
Rule. — Subtract the root of the whole number from the root of the next 
higher number, and multiply the remainder by the given decimal ; then 
the product, added to the root of the given whole number, will give the 
answer correctly to three places of decimals in the square root, and to 
seven in the cube root. 

Example. — What is the square-root of 11*14? The square-root of 
11 is 3*31 66, and the square-root of the next higher number, 12, is 
3*4641 ; the former fi'om the latter, the remainder is 0-1475, and this by 
0-14 equals 0-02065. This added to 3-3166, the sum, 3-33725, is the 
square root of 11 -14. 

To find the roots of decimals by the use of the table. Pule. — Seek for 
the gi^'en decimal in the column of numbers, and oppo.-.ite in the col- 
umns of roots will be found the answer, correct as to the figures, but re- 
quiring the decimal point to be shifted. The transposition of the deci- 
mal point is to be performed thus : For every place the decimal point is 
removed in the root, remove it in the number tioo places for the square 
root and three places for the cube root. 



APPENDIX. 27 

Examples. — By the table, the square root of 86*0 is 9*2'736, conse- 
quently, by the rule the square root of 0*86 is 0-92'736. The square 
root of 9* is 3-, hence the square root of 0*09 is 0*3. For the square 
root of 0-0657 we have 0*25632; found opposite No. 657. So, also, 
the square root of 0*000927 is 0*030446, found opposite No. 927. And 
the square root of 8*73 (whole number with decimals) is 2*9546, found 
opposite No. 873. The cube root of 0*8 is 0*928, found at No. 800 ; 
the cube root of 0*08 is 0*4308, found opposite No. 80, and the cube 
root of 0*008 is 0*2, as 2*0 is the cube root of 8*0. So also the cube 
root of 0-047 is 0*36088, found opposite No. 47. 



KULES FOE THE KEDUCTIOIN" OF DECIMALS. 

Tq reduce a fraction to its equivalent decimal. Rule. — Divide the 
immerator by the denominator, annexing cyphers as required. 

Exartiple. — What is the decimal of a foot equivalent to 3 inches ? 
3 inches is j*^- of a foot, therefore : 
3- . . . 12) 3-00 

•25 Ans. 
Another example. — What is the equivalent decimal of |- of an inch? 
I ... 9) 7-000 



•875 Ans. 

To reduce a com'pound fraction to its equivalent decimal. Rule. — In 
accordance with the preceding rule, reduce each fraction, commencing 
at the lowest, to the decimal of the next higher denomination, to which 
add the numerator of the next higher fi-action, and reduce the sum to 
the decimal of the next higher denomination, and so proceed to the last ; 
and the final product will be the answer. 

Example. — What is the decimal of a foot equivalent to 5 inches, | 
and y'g of an inch. 

The fractions in this case are, \ of an eighth, | of an inch, and ^2 of 
1 foot, therefore : 



a • • . 


. . . ^y ^ V, 






•5 






3- 


eighths. 


l... 


... 8) 3-5000 






•4375 






5- 


inches. 


h" 


. , . 12) 5-437500 






•453125 Ans. 



APPENDIX. 



The process may be condensed, thus ; write the nmnerators of th.e 
given fractions, from the least to the greatest, under each other, and 
place each denominator to the left of its numerator, thus : 



i 



12 



1-0 



3-5000 



5-487500 



•453125 Ans. 

To reduce a decimal to its equivalent in terms of lower denominations. 
Rule. — Multiply the given decimal by the number of parts in the next 
less denomination, and point off from the product as many figures to 
the right hand, as there are in the given decimal ; then multiply the 
figures pointed ofif, by the number of parts in the next lower denomina- 
tion, and point off as before, and so proceed to the end ; then the seve- 
ral figures pointed off to the left will be the answer. 

Example. — What is the expression in inches of 0'390625 feet? 
Feet 0-390625 

12 inches in a foot. 



Inches 4-687500 



8 eiofhths in an inch. 



Eighths 5-5000 



sixteenths in an eighth. 



Sixteenth 1*0 

Ans., 4 inches, f and y^g. 
Another example. — What is the expression, in fractions of an iach, 
of 0-6875 inches? 

Inches 0-6875 

8 eighths in an inch. 



Eighths 5-5000 

2 sixteenths in an eio-hth. 



Sixteenth 1-0 



Ans,, f and -,^^ 



TABLE OF CIRCLES. 

(From Gregory's Mathematics.) 

From this table may bs found by inspeotioxT the area or circamfe- 
j^nce of a circle of any diameter, and the side of a square equal to the 
area of any given circle from 1 to 100 inches, feet, yards, miles, d:c. 
If the given diameter is in inches, the area, circumference, &c., set 
opposite, will be inches ; if in feet, then feet, &c. 



? 






Side of 








Side of 


Diaci. 
•25 


Area. 


Circum. 


equal sq. 


Diam. 


Area. 


Circum. 


equal sq. 


•04908 


•78539 


•22155 


•75 


9076257 


33-77212 


9-5-2693 


•5 


•19635 


1-.57079 


-44311 


11- 


95-03317 


34-55751 


9-74849 


•75 


•44178 


2-"35619 


•66467 


•25 


99-40195 


35-34-291 


9-97005 


1- 


•7S539 


3-14159 


•886-22 


•5 


103-86890 


36-12331 


10-19160 


•25 


1-2-2718 


3-92699 


1-1077S 


•75 


108-43403 


36-91371 


10-41316 


•5 


1^76714 


4-71-238 


1-32934 


12^ 


11309733 


37-69911 


10-63472 


•75 


2-40528 


5-49773 


1-55J81; 


•25 


llr85881 


38-48451 


10-856-27 


2- 


3-14159 


6-28318 


1-77215 


•5 


1-22-71846 


39-26990 


11-07783 


•25 


3-S7fi07 


7-06358 


1-99401 


•75 


l-27^676-28 


4005530 


11-29939 


•5 


4-y0873 


7-85393 


2^21556 


13- 


132^73228 


40-84070 


11 -52095 


•75 


5-93957 


8-63937 


2-43712 


•25 


137^88646 


41-62610 


11 •74-250 


3^ 


7-0C85S 


9-4-2477 


2-6536Si 


•5 


14313881 


42-41150 


11-96406 


•25 


8-29576 


10-21017 


2-88023 


•75 


148^48934 


43-19689 


12^1 8562 


•5 


9-62112 


10-99557 


3-10179 


14^ 


153-93804 


43-98229 


12-40717 


•75 


11-04466 


11-78097 


3-3-2335 


•25 


159-48491 


44-76769 


12-62373 


4- 


12-56637 


12-55637 


3-54490 


•5 


165-12996 


45-55309 


12-35029 


•25 


14-18625 


13-35176 


3-76646 


•75 


170-87318 


46-33349 


13-07184 


•5 


15-90431 


14-13716 


3-988021 


15^ 


176-71458 


47-12338 


13^29340 


•75 


17-7-2054 


14-92256 


4-20957! 


•25 


182-65418 


47-90923 


13^51496 


5- 


19-63195 


15-70796 


4-431 13j 




188-69190 


48-69468 


13-73651 


•25 


21-64753 


] 6-49336 


4-652691 


"75 


194-82783 


49-48003 


13-95307 


•5 


23-75329 


17^27875 


4-87424 


16- 


201-06192 


50-26548 


14-17963 


•75 


25-96722 


13-06415 


5-09580J 


•25 


207-39420 


51-05088 


14-10118 


6^ 


28-27433 


18-84955 


5-317361 


•5 


213-82464 


51-83627 


14-62274 


•25 


30-67961 


19-63495 


5-53891] 


•75 


220-35327 


52-62167 


14-84430 


•5 


38^ 18307 


20-42035 


5-76047i 


17^ 


2-26-98006 


53-40707 


15^06535 


•75 


35-78470 


21-20575 


5-982031 


•25 


233-70504 


54-19247 


15^28741 


7' 


33-18456 


21-99114 


6-20353 


•5 


240-5-2818 


54-97787 


15-50897 


25 


41-28219 


2?-77654 


6-4-2514 


•75 


247-44950 


55 76326 


15-73052 


.R 


41-17S64 


23-56194 


6-64670 


18^ 


264-46900 


56-54866 


15-95208 


•75 


47-17297 


24-34734 


6-86325 
7-08981 


•25 


266-53667 


57-33406 


16-17364 


8^ 


50-26548 


25-13274 


•5 


268-80252 


58^11946 


1639519 


25 


53-45616 


2')-9IS13 


7-31137 


•75 


276-11654 


53-90486 


16-61675 


•5 


5'r74501 


26-70353 


7-53292 


19^ 


283-5-2873 


59-69026 


16-83831 


•75 


60 13204 


27-488J3 


7-75448 


•25 


291-03910 


60-47565 


17-05936 


9- 


63-61725 


28 27433 


7-976J4I 


•5 


298-64765 


61-26105 


17-28142 


•25 


(;7-20063 


29-05973 


8-197591 


•75 


306-35437 


62-04645 


17-50298 


•5 


70-8 S2 18 


29-84513 


8-41915 


20^ 


314 •159-26 


62^83185 


17-72453 


•75 


74-6G191 


30-63052 


8-64071 


•25 


322-06233 


6361725 


17-94609 


10^ 


78-53981 


31-41592 


8 86-226 


•5 


330^06357 


64^40264 


18-16765 


•25 


82-51539 


32-20132 


9-08382 


•75 


338- 16-299 


65 •18804 


18-38920 


•5 1 


86-59014 


32-98672 


9-30533; 


21- 1 


346^36059 


65-97344 


18-61076 



30 



JLPPENDIX. 









Side of 








Side of 
eoual sq. 


Diam. 


Area. 


Circum. | 


equal sq. 


Diam. 


Area. 


Circum. 


Ti^ 


354^65635 


66^75334 


18-83-?.32 


38- 


1134- 11494 


119-38052 


33-67662 


5 


363^05030 


67-54424 


19-05387 


-25 


1149-08660 


120-16591 


33-89317 


•75 


37 1^5424 1 


6S-329G4 


19-27543 


-5 


1164-15642 


1-20-95131 


34-11973 


22- 


330^13271 


69-115U3 


19-49699 


•75 


1179-32442 


121-73671 


34-341-29 


•25 


388-82117 


69-90043 


19-71854 


39- 


1194-59060 


1-22-52211 


34-56285 


•5 


357 60782 


70-63583 


19-9401() 


•25 


1209-95495 


1-23-30751 


34 78440 


•75 


406-49263 


71-471-23 


20-16166 


•5 


1-2-25-41748 


124-09290 


35-00596 


23- 


415-475B2 


72-25663 


20-38321 


•75 


15^10-97813 


1-24-87830 


35-2275-i 


•25 


424-55679 


73-04-202 


20-60477 


40- 


125')-63704 


125-66370 


35-44907 


•5 


433-73613 


73-82742 


20-82633 


•25 


1-272-3941] 


128-44910 


35-67063 


•75 


443-01365 


74-61-282 


2104783 


•5 


1288-24933 


127-23450 


35-89219 


24^ 


452-38934 


75-388-22 


21-26944 


•75 


1304-20273 


128-01990 


36-11374 


•25 


461-86320 


76-18362 


21-49100 


41^ 


1320-25431 


123-805-29 


36-33530 


•5 


471-435-24 


76-96902 


21-71255 


•25 


1336-40406 


129-59060 


36-55686 


•75 


481-10546 


77-75441 


21-93411 


•5 


1352-65198 


130-37609 


36-77341 


25- 


490-87385 


78-53981 


22-15567 


•75 


1368-99808 


131-16149 


.36-99997 


•25 


500-74041 


79-32521 


22-37722 


42^ 


1335-44236 


131-94689 


37-22153 


•5 


510-70515 


80-11061 


22-59878 


-25 


1401-98480 


132-73-228 


37--44308 


•75 


520-76306 


80-89601 


22-82034 


•5 


1418-62543 


133 51763 


37 66464 


26- 


530-9-^915 


81-63140 


23-04190 


-75 


1435-36423 


131-30308 


37-83620 


•25 


541-18342 


82-46fiS0 


23-26345 


43- 


1452-20120 


135-08348 


33-10775 


•5 


551-54586 


83-25-220 


23-48501 


•25 


1469-13635 


135-87333 


38-3-2931 


•75 


552^00147 


84-03760 


23 70657 


•5 


1486-16967 


13G-65928 


38-5.5037' 


27- 


572-55526 


84-82300 


23-9-2812 


-75 


1503-30117 


137-44467 


33-77242 


•25 


583-20722 


85-60839 


24 14968 


44- 


1520-53081 


133-23007 


33-993.8 


•5 


593-95736 


86-39379 


24-37124 


-25 


1537-8586:> 


1P9-01547 


39-21554 


•75 


604-80567 


87-17919 


24-59279 


•5 


1556-23471 


139-800H7 


39-43709 


28^ 


615-75216 


87-96459 


24-31435 


-75 


1572-803i'U 


140-53627 


3965S65 


•25 


626-79682 


88-74999 


25-03591 


45- 


15i?043]23 


141-37166 


39-83021 


•5 


637-93965 


89^53539 


25-25746 


•25 


1608-15182 


142-15706 
142 94-246 


40-10176 


•75 


649-13066 


90-32078 


25-47902 


-5 


1625-97054 


40-32332 


29^ 


660-51935 


91-10613 


25-70053 


•75 


1W3 83744 


143-72738 


4054488 


•25 


671-95721 


91-89153 


25-92213 


46- 


1661-90-251 


144 513-26 


40-76643 


•5 


683-49275 


92^67698 


26-14369 


•25 


16S0-01575 


145-29866 


40-93799 


•75 


695-12646 


93-4623S 


26-36525 


•5 


1698-22717 


146-08405 


41-20955 


33- 


706^85834 


94^24777 


26-58680 


•75 


1716-53677 


146-86945 


41^43110 


•25 


718-68840 


95-03317 


25-80836 


47- 


1734-9445-1 


147-65435 


41^65266' 


•5 


730-61664 


95-81857 


27-02992 


•25 


1753-45048 


148-44025 


41-87422| 


•75 


74264305 


96^60397 


27-25147 


•5 


1772-05460 


149-2-2565 


42-095771 


31- 


751-76763 


97-38937 


27-47303 


•75 


1790-75639 


150-01104 


42-31733 


•25 


766-99039* 


98-17477 


27-69459 


48^ 


1809 557:^6 


150^79644 


42-53389 


•5 


779-31132 


98 S6016 


27-91614 


•25 


1828-45001 


151-53184 


42-76044 


•75 


791-73043 


99-74556 


23-13770 


•5 


1847-45232 


152-36724 


42-98200 


32- 


804-24771 


100-53096 


23-^15926 


•75 


1866-54782 


153-15-264 


43-20356 


•25 


816-86317 


101-31636 


-2S -58081 


49- 


1885-74099 


153-93804 


43 4-2511 


•5 


8-29-57681 


102-10176 


28-h0237 


•25 


1905-83233 


154-72343 


43-64667 


•75 


812-38861 


10-2-83715 


29 U2393 


•5 


1924-42184 


155-50383 


43-86823 


33^ 


855-29859 


103C7255 


. 29-24548' 


•75 


1943-90954 


156-29423 


44-U897S 


•25 


868-30675 


104-45795 


29-46704 


50^ 


1963-49540 


157-07963 


44-31134 


•5 


831^41308 


105-24335 


2 J -68860 


•25 


1983-17944 


157-96503 


44-53290 


•75 


894-61759 


106-02375 


29-9 1015^ 


•5 


2C02-96166 


158-65042 


44-75445 


34- 


907^92027 


106-81415 


3013171 i 


•75 


2J-22-84205 


159-43582 


44-97601 


25 


92 132 113 


107-59954 


30-35327; 


51^ 


2042-82062 


160-22122 


45-19757 


•5 


934-82016 


108-33494 


30-57482! 


•25 


2062-89736 


161-0(662 


45-41912 


•75 


948 41736 


109-17034 


30-79638' 


•5 


2083-07227 


161-79202 


45-64068 


35- 


96211275 


109-95574 


31-01794 


•75 


2103-31536 


162-57741 


45-852-24 


•25 


975^90630 


110-74114 


31 •23949; 


52- 


21-23-71663 


163-36281 


46-0gl30 


•5 


989-79803 


111-5-2653 


3r46105' 


•25 


2144-18607 


164-14821 


46-30535 


•75 


1003-78794 


112-31193 


31-68-261 


•5 


2164-75368 


164-93361 


46 5-2691 


36- 


1017-87601 


113-09733 


3r90416 


•75 


2185-41947 


165-71901 


46-71847 


•25 


1032-06227 


113-88-273 


32^ 125721 


53^ 


2206-18344 


166-50441 


46-9700-2 


•5 


1046-34670 


114-66313 


32-34728! 


•25 


2227-04557 


167-2S930 


47-19153 


•75 


1060-72930 


115-45353 


32-55383| 


-5 


2248-00589 


168-075-20 


47-41314 


37- 


1075-21003 


116-238:'2 


32-79039 i 


-75 


2269-06433 


168-88060 


47-63469 


•25 


1089-78903 


117-0'243--i 


33-01195; 


54- 


2-290-22104 


169-64600 


47-85625 


•5 


1104-46616 


117-80972 


33-23350 


•25 


2311-47538 


170-43140 


43-0778-1 


•75 


1119-2414'- 


! 118-59572 


33-455061 


•5 


2332-8-2889 


171-21679 


48-29936 



APPENDIX. 



31 









Side of 








Side of 


Diam. 


Area. 


CircuiTi. 


equal sq. 


Diam. 


Area. 


Circum. 


equal sq. 


"54^ 


2354 -281^)8 


172-00219 


48-52092 


71-5 


"lois^m 


2-24-62337 


63-36522 


55- 


2375^82-)44 


172-78759 


48-74248 


•75 


4043-27883 


225-40927 


63-58678 


•25 


2397^47698 


173-57299 


48-96403 


72- 


4071 •50407 


226-19467 


63-80333 


•5 


2419-2^2269 


174-3583.) 


49-18559 


•25 


4099-82750 


226-98006 


64-02989 


•75 


2UrC6657 


175-14379 


49-40715 


•5 


4128-24909 


227-76546 


64-25145 


53- 


24<-)3-00864 


175-92918 


49-62870 


•75 


4156-76886 


228-55086 


64-47300 


•25 


2J85'04887 


176-71453 


49-850-26 


73- 


4185-33681 


229-33626 


64-69456 


•5 


2507^ 18728 


177-49998 


50-07182 


•25 


4214-10293 


230-12166 


64-91612 


■75 


2520^4-J337 


178-28538 


50-29337 


•5 


4242-91722 


230^90706 


65-13767 


57- 


2551-75863 


179-07<»78 


50-51493 


•75 


4271-82969 


23b69-245 


65-35923 


•25 


2574-19156 


179-85617 


50-73649 


74^ 


4300-84034 


232-47785 


65-58079 


•5 


2596-72267 


180-64157 


50-95304 


•25 


4329-94916 


233-26325 


65-80234 


•75 


2619-35196 


181-426^7 


51-17960 


•5 


4359-15615 


234-04865 


66-02390 


53^ 


2642-07942 


182 21237 


51-40116 


•75 


4388-46132 


234-83405 


66-24546 


•25 


2664 90505 


182-99777 


51-6-2271 


75^ 


4417-86465 


235-61944 


66^46701 


•5 


2637-82336 


183-78317 


51 84427 


•25 


4447-36618 


236-40484 


66-68857 


•75 


27] 0-85084 


184-5685S 


52-06583 


•5 


4476-965^8 


237-19024 


66-91043 


59- 


2733-97100 


185-353v6 


52-28738 


•75 


4506-66374 


237-97564 


67-13168 


•25 


2757-18933 


18613936 


52-50894 


76^ 


4536-45979 


238-76104 


67-35324 


•5 


2780-50584 


186 •9-2476 


52-73050 


•25 


4566 35400 


239-54643 


67-57480 


•75 


23039-2053 


187-71016 


52-95205 


•5 


45;;6 34640 


240-33183 


67-79635 


6d- " 


2S27-43338 


183-49555 


53-17364 


•75 


4626-43696 


241-11723 


6801791 


•25 


2351-04442 


189-28095 


53 39517 


77^-' 


46536-2571 


241-90263 


68-23947 


■5 


2874-75362 


190-06635 


53 61672 


•25 


4636-91-262 


242-68803 


68-46102 


•75 


2893-56100 


19)-85175 


53-83328 


•5 


4717-29771 


243-47.343 


68-68258 


61- 


2922-46656 


191'637]5 


51-05984 


•75 


4747-78098 


244-25382 


68-90414 


•25 


2946-47029 


192-4-2255 


54-28133 


78^ 


4778-36-242 


245 04422 


69-1-2570 


•5 


■ 2-^70-57220 


193-20794 


54-50295 


•25 


4809-04204 


245-82962 


6934725 


•75 


2.MJ4-77223 


193-99334 


54-72451 


•5 


4839-81983 


246-61502 


69-56881 


02- 


3019-07054 


194-77374 


54-94606 


•75 


4870-79579 


247-40042 


69-79037 


•25 


3a43-4e697 


195^56414 


55-16762 


79- 


4901-66993 


248-18581 


70-01192 


•5 


3J67-96157 


] 96-34954 


55-33918 


•25 


4932-74225 


248-97121 


70-23348 


•75 


3092-55435 


197-13493 


55-61073 


•5 


4963-91274 


249-75661 


70^45504 


63- 


3117-24531 


197-9-2033 


55-83229 


•75 


4995-18140 


250-34201 


70-67659 


•25 


3142 03444 


198^70573 


53-053S5 


89^ 


5026-54824 


251-32741 


70-89815 


•5_ 


3166-92174 


199-49113 


56-27540 


•25 


505801 3-25 


252-11-281 


71-11971 


•75 


3191-90722 


200-27653 


56-49696 


•5 


5089-57644 


252-898-20 


71-34126 


04 • 


3216-99087 


201-06192 


56-71852 


•75 


5121-23781 


253-63360 


71-56282 


•25 


3242-17270 


201-84732 


56-94007 


sr 


5152-99735 


254-46900 


71^78438 


•5 


3267-45270 


20263272 


57-16163 


•25 


5184-85506 


255-25440 


72-00593 


•~5 


3292 83083 


203-41812 


57-38319 


•5 


5216-81095 


256-03989 


72-22749 


55- 


3318-307-24 


•204-20352 


57-60475 


•75 


5248-86501 


256-82579 


72-44905 


•25 


3343-88176 


204-98892 


57-82630 


82^ 


5281 -017-25 


257-61059 


72^67060 


•5 


3369-55447 


205-77431 


5304786 


•25 


531326766 


258-39599 


72^89216 


•75 


3395-32534 


206-55971 


53-26942 


•5 


5345-61624 


259-18139 


73 11372 


66^ 


3421 -19439 


207-34511 


53-49097 


•75 


5378-06301 


259 96679 


73-33527 


25 


344716162 


203-13051 


58-71253 


83^ 


5410-60794 


260-75219 


73-55683 


■5 


3473-22702 


203-91591 


58-93409 


•25 


5443-25105 


261-53753 


7377839 


•75 


3499 39060 


209-70130 


59-15564 


•5 


5475-99234 


262-32-298 


73-99994 


G7^ 


3525-65235 


210 48rt70 


59-37720 


•75 


5508-83180 


263-10833 


74-22150 


25 


3552-0r228 


211-27210 


59-59876 


84 • 


5541-76944 


263-89378 


74-44306 


•5 


3573-47033 


212^05750 


59-82031 


•25 


5574-80525 


264-67913 


74-66461 


•75 


360502665 


212-84290 


6004187 


•5 


5607-93923 


265-46457 


74-88617 


68- 


3631-68110 


213-62-S30 


60-26343 


•75 


5641-17139 


266-24997 


75-10773 


•25 


3658-43373 


214-41369 


60-43498 


85^ 


5674-50173 


267-03537 


75-32928 


•5 


3685-284 53 


215-19909 


60-70654 


•25 


5707-93023 


2G7^8-2077 


75-55084 


•75 


3712-23350 


215-98449 


60-923 lu 


•5 


5741-45692 


268-60617 


75-77240 


69- 


3^39-28065 


216 76989 


6M4y65 


•75 


5775-08178 


269-39157 


75-99395 


•25 


3766-42597 


217-555-29 


61-37121 


86^ 


5808-80481 


270-17696 


76-21551 


•5 


3793-66947 


21834068 


61-59277 


•25 


5342-62602 


270-96236 


76-43707 


■75 


332101115 


21i:- 12608 


61-81432 


•5 


5876-54540 


271-74776 


76-65362 


70 


3348-45100 


219-91 143 


62-03533 


•75 


5.) 10-56296 


272-53316 


76-88018 


25 


3875-98902 


220-69(583 


62-25744 


87^ 


594467S69 


273-31856 


7710174 


•5 


3.i03^625-22 


'2^1-48-2-28 


62^47899 


•25 


5978-89260 


274-103J5 


77-32329 


•75 


3931-35959 


222-2;v/68 


62-70055 


•5 


6013-20468 


274-88935 


77-54485 


71 


3959^192]4 


223-05307 


62-92211 


•75 


604761494 


275-67475 


77-76641 


•2'- 


3987-12286 


22363347 


6314366 


88- 


6082-12337 


276-46015 


77-98796 



32 



APPENDIX. 









Side of 1 1 






Side oi 


Diam. 


Area. 


Circum. 


equal sq. 


Diam. | 


Area. 


Circum. 


equal sq. 


Td^ 


6116-72993 


277-24555 


78-20952 


"94^1" 


6976-74097 


296-09510 


83-52683 


•5 


6151-43476 


278-03094 


73-43103 


•5 1 


7012-80194 


296-88050 


83-74344 


•75 


6186-23772 


278-81634 


78-65263 


•75 


7050-96109 


297-66590 


83-97000 


89- 


622M3S85 


279-60174 


78-87419 


95- i 


7083-21342 


298-45130 


84-19155 


•25 


6-256^13315 


230-33714 


79-09575 


-25; 


7 125 •57-992 


299-23670 


84-4131 1 


•5 


6291-23563 


231-17254 


79-31730 


-5 i 


7163-02759 


30002209 


84-63467 


•75 


6326-43129 


281-95794 


79-53336 


•75; 


7-200-57944 


300-80749 


84-85622 


90- 


6361-72512 


282-74333 


79-76042 


96- 1 


7238-22947 


301-59239 


85-07778 


•25 


6397-11712 


233-5-2373 


79-98198 


-25 


7275-97767 


302-37829 


85-29934 


•5 


6432-60730 


284-31413 


80-20353 


•5 i 


7313-82404 


303-16369 


85-52089 


•75 


6468-19566 


285-09953 


80-42509 


-75! 


7351-76859 


303-94908 


85-74245 


91- 


6503-83219 


285-88493 
286-67032 


80-64669 


97- I 


7339-81131 


304-73448 


85-96401* 


.25 


6533-66689 


80-85820 


-25 


7427-95221 


305-51988 


86-18556 


•5 


6575-51977 


287-45572 


81-03976 


•5 1 


7466-191-29 


306-30528 


8n-40712 


•75 


6611-53082 


288-24112 


81-31132 


-75 1 


7504-52853 


307-09068 


86-6-236S 


92^ 


6347-61005 


289-02652 


81-53287 


93- j 


7542-96396 


307-87603 


86-85023 


•25 


6683-78745 


289-81192 


81-75443 


•25: 


7581-49755 


308-66147 


87-07179 


'5 


6720-06303 


290-59732 


81-97599 


■5 1 


7620-12933 


309-44687 


87-29335 


•75 


6756-43678 


291-33271 


82-19754 


•75, 


7653-859-27 


310-232-27 


87-51490 


93- 


6792-90871 


292-16811 


82-41910 


99^ i 


7697-68739 


311-01767 


87-73646 


•25 


68-29-47881 


292-95351 


82-04066 


•25 


7736-61369 


311-80307 


87-95802 


•5 


6866-14709 


293-73391 


82-86221 


•5 


7775-63316 


312-58846 


88^ 17957 


•75 


6902-91354 


294-52431 


83-08377 


•75 


7814-76081 


313^37336 


88-40113 


94- 


6939-77817 


295-30970 


83-30533 II 100- 'j 


7353-93163 


314^15926 


88^62289 



The following rules are for extending the use of the above table. 

To find the area, circumference, or side of equal square, of a circie 
having a diameter of more than 100 inches, feet, SfC. Rule. — Divide 
the given diameter by a number that will give a quotient equal to some 
one of the diameters in the table ; then the circumference or side of 
equal square, opposite that diameter, multiplied by that divisor, or, the 
area opposite that diameter, multiplied by the square of the aforesaid 
divisor, will give the answer. 

Exarajple. — What is the circumference of a circle whose d'ameter is 
228 feet 1 228, divided by 3, gives 76, a diameter of the table, the cir- 
cum.ference of which is 238^761, therefore : 
238^7G1 
3 



716^283 feet. Ans. 
Another example. — What is the area of a circle liaving a diameter 
of 150 inches ? 150, divided by 10, gives 15, one of the diameters in 
the table, the area of which is 176-71458, therefore : 
176-71458 

100= 10 X 10 



17,671^45800 inches. Ans. 
To find the area, circumference, or side of equal, square, of a circle 
having an intermediate diameter to those in the tahle. Rule. — Multiply 
the given diameter by a number that will give a product equal to some 
one of the diameters in the table ; then the circumference or side of 
equal square opposite that diameter, divided by that multiplier, or, the 
area opposite that diameter diviled by thi rrquare of the aforesaid mul 
tiplier, will give the answer. 



APPENDIX. 



33 



Example. — What is the circumference of a circle wliose diameter is 
6j^, or 6-125 inches ? 6* 125, multiplied by 2, ^ives 12-25, one of th*> 
■diameters of the tabh;, whose circumference is 8S-484, therefore : 
2)38-484 



19-242 inches. Ans. 
Anomcr example. — What is the area of a circle, the diameter of 
which is 3-2 feet '? 8-2, multiplied by 5, gives 16, and the area of 16 
is 201-0619, therefore : 

5 X 5 — 25)201-0619(8-0424 + feet. Ans. 
200 

106 
100 

61 
50 

119 
100 

19 

- Note. — The diameter of a circle, multiplied by .3-14159, will give 
its circumference ; the square of the diameter, multiplied by -78539, 
will give its area ; and the diameter, multiplied by -88622, will aive 
the side of a square equal to the area of the circle. 



TABLE SHOWING THE CAPACITY OF WELLS, CISTERNS, &C. 

The gallon of the State of New York, by an act passed April 11, 1S51, is required to conform 
to the standard gallon of the United States government. This standard gallon contains 231 cubic 
inches. In conformity with this standard the following table has been computed. 



One foot in depth of a cistern of 
3 feet diameter will contain 



4 




41 
5 


ii 


5* 
6 




i 




8 


a 


9 


u 


10 


u 


12 


u 



52-812 


gallons 


71-965 


u 


93-995 


a 


118-963 


u 


146-868 


" 


177-710 


u 


211-490 


u 


248-207 


li 


287-861 


a 


375-982 


" 


475-852 


" 


587-472 


u 


845-0."0 


'{ 



To reduce cubic .eet to gall )i)s, multiply by 7'48. 



TABLE OF POLYGONS. 

(From Gregory's Matlieinatics.) 



- t 

a-3 




3Iulti pliers for 


Railiiis of oir- 


Fact(!rs for 


S-^ 




areas. 


cura. circle. 


sides. 


3 


I'rigon 


0-433v0127 


0-5773503 


1 ■73-2051 


4 


Tetragon, or Square 


1-0000000 


0-7071068 


1-414214 


5 


Pentagon - 


1-7204774 


0-8506508 


1-175570 


6 


Hexagon 


2-5980762 


1-0000000 


1-000000 


7 


Heptagon - 


3-6339124 


1-1523824 


0-867767 


8 


Octagon 


4-8284271 


1-3085628 


0-765367 


9 


Nonagon - 


6-1818242 


1-4619022 


0-684040 


10 


Decagon 


7-6942088 


1-6180340 


0-618034 


11 


Undecagon 


9-3656399 


1-7747324 


0-563465 


|12 


Dodecagon - 


11-1961524 


1-9318517 


0-517638 



To find the area of any regular polygon, loliose sides do not exceed 
twelve. Rule. — Multiply the square of a side of the given polygon by 
the number in the column termed Multipliers for areas., standing op 
posite the name of the given polygon, and the product will be the an- 
swer. Example. — What is the area of a regular heptagon, whose 
sides measure each 2 feet ? 

3-6339124 

4 = 2x2 



14-5356496: Ans. 
To find the radius of a circle which will circumscribe any regular 
polygon given, lohose sides do not exceed twelve. Rule. — Multiply a 
side of the given polygon by the number in the column termed Radius 
of circumscrihing circle, standing opposite the name of the given poly- 
gon, and the product will give the answer. Example. — What is the 
radius of a circle which will circumscribe a regular pentagon, whose 
sides measure each 10 feet ? 

•8506508 
10 



8-5065080: Ans. 
To find the side of any regular polygon that raay he inscribed within 
a given circle. Rule. — Multiply the radius of the given circle by the 
number in the column termed Factors for sides, standing opposite the 
name of the given polygon, and the product will be the answer. Ex- 
ample. — What is the side of a regular octagon thtt may be inscribed 
within a circle, whose radius is 5 feet ? 
•765367 
5 



3-8-26835: Ans. 



WEIGHT OF MATERIAI^. 



Woods, 


lbs. in a 
cubic foot. 


Metals. cubic foot. 


Apple, - 


- 49 


Wire-drawn brass, - 534 


Ash, 


45 


Cast brass, - - 506 


Beach, - - - 


- 40 


Sheet-copper, - - 549 


Birch, 


45 


Pure cast gold, - - 1210 


Box, 


- 60 


Bar-iron, - 475 to 487 


Cedar, 


28 


Cast iron, - . 450 to 475 


Virginian red cedar. 


. 40 


Milled lead, - - - 713 


Cherry, 


38 


Cast lead, - - 709 


Sweet chestnut, 


- 36 


Pewter, - - - 453 


Horse-chestnut, - 


34 


Pure platina, - - 1345 


Cork, 


- 15 


Pure cast silver, - - 654 


Cypress, 


-. 28 


Steel, - - 486 to 490 


Ebony, - 


- 83 


Tin, - - - - 456 


Elder, 


43 


Zinc, - - - 439 


Elm, 


- 34 


Stone, Earths, ^c. 


Fir, (white spruce,) 


29 


Brick, Phila. stretchers, 105 


Hickory, 


- 52 


North river common hard 


Lance-wood, 


59 


brick, - - - 107 


Larch, - 


- 31 


Do. salmon brick, 100 


Larch, (whitewood,) 


22 


Brickwork, about - 95 


Lignum-vitse, - 


- 83 


Cast Roman cement, - 100 


Logwood, 


57 


Do. and sand in equal parts, 113 


St. Domingo mahogany, - 45 


Chalk, - 144 to 166 


Honduras, or bay mahogany, 35 


Clay, - - - - 11& 


Maple, 


47 


Potter's clay, - 112 to 130 


White oak, 


43 to 53 


Common earth, 95 to 124 


Canadian oak. 


54 


Flint, - - - - 163 


Red oak, 


- 47 


Plate-glass, - - 172 


Live oak, 


76 


Crown-glass, - - - 157 


White pine, - 


23 to 30 


Granite, - - 158 to 187 


Yellow pine. 


34 to 44 


Q-uincy granite, - - 166 


Pitch pine. 


46 to 58 


Gravel, - - - 109 


Poplar, 


25 


Grindstone, - - - 134 


Sycamore, 


- 36 


Gvpsuni, (Plaster-stone,) 142 


Walnut, 


40 


Unslaked lime, - - 52 



65 



36 



APPENDIX. 



lbs. 


vna 




Jb8.ina 


cubic foot. 




cnbic/ooL 


Limestone, - - 118 to 198 


Common blue stone. 


160 


Marble, - - 161 to 177 


Silver-gray flagging. 


- 185 


New mortar, - 


107 


Stonework, about. 


120 


Dry mortar. 


90 


Common plain tiles. 


- 115 


Mortar with hair, (Plaster- 




Sundries. 




ing,) - 


105 


Atmospheric air. 


- 0-075 


Do. dry, 


86 


Yellow beeswax, - 


- 60 


Do. do. including lath 




Birch-charcoal, - 


34 


and nails, from 7 to 11 




Oak-charcoal, 


- 21 


lbs. per superficial foot. 




Pine-charcoal, - 


17 


Crystallized quartz. 


165 


Solid gunpowder, - 


- 109 


Pure quartz-sand. 


171 


Shaken gunpowder. 


58 


Clean and coarse sand, 


100 


Honey, 


- 90 


Welsh slate, - 


180 


Milk, 


64 


Paving stone. 


151 


Pitch, - 


- 71 


Pumice stone. 


56 


Sea-water, 


64 


Nyack brown stone, - 


148 


Rain-water, - 


- 62-5 


Connecticut brown stone. 


170 


Snow, 


8 


Tarrjnown blue stone, - 


171 


Wood-ashes, 


59 



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